- Email: [email protected]
Gravity and electromagnetism on conic sedenions
Applied Mathematics and Computation 188 (2007) 948–953 www.elsevier.com/locate/amc
Gravity and electromagnetism on conic sedenions Jens Ko¨plinger 406 Aberdeen Ter, Greensboro, NC 27403, USA
Abstract Conic sedenions from C. Muse`s’ hypernumbers program are able to express the Dirac equation in physics through their hyperbolic subalgebra, together with a counterpart on circular geometry that has earlier been proposed for description of gravity. An electromagnetic field will now be added to this formulation and shown to be equivalent to current description in physics. With use of an invariant hypernumber modulus condition, a description of quantum gravity with field will be derived. The resulting geometry reduces in very good approximation to relations expressible through customary tensor algebra. However, deviations are apparent at extreme energy levels, as shortly after the Big Bang, that require genuine conic sedenion arithmetic for their correct description. This is offered as method for exploration into bound quantum states, which are not directly observable in the experiment at this time. Extendibility of the invariant modulus condition to higher hypernumber levels promises mathematical flexibility beyond gravity and electromagnetism. 2006 Elsevier Inc. All rights reserved. Keywords: Hypernumbers; Quantum gravity; Electromagnetism, Conic complex numbers; Countercomplex numbers; Sedenions; Octonions; Physics on hyperbolic and circular geometry
1. Introduction In order to qualify and sufficiently support a quantum theory of gravitation on genuine conic sedenion arithmetic (from C. Muse`s’ hypernumbers program), it was first shown that the Dirac equation in physics is expressible on hyperbolic octonion arithmetic [1]. Rotation in the (1, i0) plane yields a counterpart on circular (Euclidean) geometry, which exhibits certain primitive properties of quantum gravity [2]. For large-body (non-quantum) physics, an alignment program was developed from an invariant modulus condition [3], and shown to be consistent with the General Relativity formalism for gravity. This paper will conclude definition of the computational framework for a proposed quantum theory of gravitation and electromagnetism on hypernumbers, by supplying a physical force field to the formulation. An electromagnetic field will be added to the hyperbolic subalgebra and shown to be equivalent to current description in physics, supporting a conjecture by analogy for the circular subalgebra to describe quantum gravitational interaction.
E-mail address: [email protected] 0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.10.050
J. Ko¨plinger / Applied Mathematics and Computation 188 (2007) 948–953
949
Certain mixing effects become apparent at extreme energies, which cannot be separated into the individual constituent forces of gravity and electromagnetism anymore. This makes conic sedenion arithmetic a needed tool for further investigation into such effects. Referring to the power orbit concept, it will be remarked that hypernumbers offer additional mathematical versatility to satisfy a generalized invariant hypernumber modulus theorem, beyond the description of gravity and electromagnetism herein. 2. Electromagnetism The Dirac equation with electromagnetic field is a fundamental building block in describing dynamic interaction of spin 1/2 particles (like electrons or protons). In a simple form it can be written as ½cð^ p eAÞ mW ¼ 0;
ð1Þ
(from [4] equation 32.1) and contains an operator p^ and certain implicit summations. In order to map this relation to conic sedenion arithmetic as in [1], it will now be written explicitely1 in terms of space-time derivatives ol. The terms c, ^ p, A, and W in (1) are summed over three indices l, m, r 2 {0, 1, 2, 3}, with implicit use of a metric tensor glm (describing Minkowski metric; glm = 0 for l5m, g00 = 1, and gii = 1 otherwise), and result in four equations now indexed with q 2 {0, 1, 2, 3}. The c are constant, anti-commuting 4 · 4 matrices [cl]qr on complex numbers, ^ p are the quantum mechanical operators for energy ^p0 io0 and momentum ^pi ioi (i 2 {1, 2, 3}), e is the electric charge of the spin 1/2 particle, m is its mass, and Am the electromagnetic field actingPupon it. Both Am and Wr are a function of time t x0 and space ~ x fx1 ; x2 ; x3 g. With Al :¼ 3m¼0 glm Am one can then write (1) in the following form: ! 3 3 X X ½cl qr glm ð^pm eAm Þ dqr m wr ¼ 0; ð2Þ r¼0
l;m¼0
3 3 X X r¼0
i½cl qr ol
½cl qr eAl
! dqr m wr ¼ 0:
ð3Þ
l¼0
Interpreting the summation terms i½cl qr ol ½cl qr eAl as coefficients ol and eAl to bases icl and cl respectively, one finds the following algebraic relations: (1) All 8 bases icl and cl, as well as, the term m (to basis ‘‘1’’ dqr) are linear independent. (2) Since all cl anti-commute (clcm = cmcl for l5m), the bases to the ol factors anti-commute, as well as the bases to the eAl . (3) The eAl bases anti-commute with bases to om factors for l5m, and commute for l = m. Conic sedenion arithmetic to basis elements bcon16 2 {1, i1, . . . , i7, i0, e1, . . . , e7} and the definitions rEM con16 :¼ ð0; o0 ; 0; 0; 0; eA3 ; eA2 ; eA1 ; 0; eA0 ; 0; 0; 0; o3 ; o2 ; o1 Þ; r i r i r i r i WEM con16 :¼ w0 ; w0 ; w1 ; w1 ; 0; 0; 0; 0; 0; 0; 0; 0; w2 ; w2 ; w3 ; w3 ; with m (m, 0, . . . , 0) allow to express (3) as EM rcon16 m WEM con16 ¼ 0:
1
ð4Þ ð5Þ
ð6Þ
Please note that physical constants c, h, and G are set to 1, and all summations will be written using lower indices only. A metric tensor will be specified if present.
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J. Ko¨plinger / Applied Mathematics and Computation 188 (2007) 948–953
Proof. For Al = 0 relation (6) reduces to the correct form of the Dirac equation without field on hyperbolic octonion algebra.2 Embedded in conic sedenions, demanding invariance under a transformation of the wave function EM i0 v WEM con16 ! Wcon16 e ;
ð7Þ
in direct analogy to its description in physics (e.g. [4] relation 32.3) with ð8Þ
ol v :¼ eAl ; yields i1 o0 ! i1 ðo0 i0 eA0 Þ ¼ i1 o0 þ e1 eA0 ;
ð9Þ
e5 o3 ! e5 ðo3 þ i0 eA3 Þ ¼ e5 o3 þ i5 eA3 ; e6 o2 ! e6 ðo2 i0 eA2 Þ ¼ e6 o2 i6 eA2 ;
ð10Þ ð11Þ
e7 o1 ! e7 ðo1 þ i0 eA1 Þ ¼ e7 o1 þ i7 eA1 :
ð12Þ
The number bases of the Al terms correspond to (4), and the conic sedenion bases {i1, e5, e6, e7} and {e1, i5, i6, i7} associated with the ol and Al terms respectively satisfy above algebraic relations of the icl and cl bases used in physics (change of sign in Al ¼ Al does not alter these relations). This concludes the proof. h 3. Gravity In an identical procedure to electromagnetism above, the circular octonionic Dirac equation proposed for the description of gravity in [2] (relation 9 there) can be extended by a field eA0 using the definitions rGr con16 :¼ ð0; o0 ; 0; 0; 0; o3 ; o2 ; o1 ; 0; eA0 ; 0; 0; 0; eA3 ; eA2 ; eA1 Þ; WGr con16
:¼
ðwr0 ; wi0 ; wr1 ; wi1 ; wr2 ; wi2 ; wr3 ; wi3 ;
0; 0; 0; 0; 0; 0; 0; 0Þ;
ð13Þ ð14Þ
to Gr ðrGr con16 mÞWcon16 ¼ 0:
This is obtained by demanding invariance of WGr con16
!
i0 v WGr con16 e ;
ð15Þ WGr con16
under the same transformation as (7) above, ð16Þ
(with identical definition of olv := eAl using electromagnetic field Al and electric charge e of the particle). Such an ansatz implies that every spin 1/2 particle with mass at rest m also has an associated electrical charge e. While this is true for most fundamental particles (electron, muon, tau, all quarks, and associated anti-particles), neutrinos are currently assumed to have a mass at rest but no electric charge. Since they only interact through gravitation and the weak force (with associated weak charge), a formalism that would describe both gravity and weak interaction could be more appropriate for these particles. They are therefore excluded from the current investigation. In order to define the computational framework for a proposed quantum theory of gravitation and electromagnetism on hypernumbers, the invariant hypernumber modulus theorem from [3] will now be proposed to be valid for quantum gravity. This theorem was concluded to be sufficiently founded for large-body (nonquantum) gravitation through the NatAliE equations program, and rotation in the (1, i0) plane appears to be a plausible way to transform proven relations for electromagnetism on hyperbolic geometry into gravity on circular geometry.
2
See [1] Eq. (5), but note that definition (4) there is incorrect and should be $hyp8 := (m, o0, 0, 0, 0, o3, o2, o1), as well as definition (3) should be Whyp8 :¼ ðwr0 ; wi0 ; wr1 ; wi1 ; wr2 ; wi2 ; wr3 ; wi3 Þ. The different definitions were the result of using a conic sedenion multiplication table which identified the classical octonion element ‘‘l’’ with sedenion element i4 instead of i4, therefore not being consistent with the cited sources [5,6].
J. Ko¨plinger / Applied Mathematics and Computation 188 (2007) 948–953
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With use of a real-number mixing angle a and the definitions :¼ ð0; o0 ; 0; 0; 0; 0; 0; 0; 0; eA0 ; 0; 0; 0; 0; 0; 0Þ; rGr;EM Q1
ð17Þ
rGr;EM Q2
ð18Þ
r
Gr;EM
WGr;EM Q1 WGr;EM Q2
:¼ ð0; 0; 0; 0; 0; o3 ; o2 ; o1 ; 0; 0; 0; 0; 0; eA3 ; eA2 ; eA1 Þ; :¼
rGr;EM Q1
:¼
ðwr0 ; wi0 ; wr1 ; wi1 ; 0; 0; 0; 0;
:¼
ð0; 0; 0; 0; wr2 ; wi2 ; wr3 ; wi3 ; WGr;EM þ expði0 aÞWGr;EM ; Q1 Q2
WGr;EM :¼
þ
expði0 aÞrGr;EM ; Q2
ð19Þ
0; 0; 0; 0; 0; 0; 0; 0Þ; 0; 0; 0; 0; 0; 0; 0; 0Þ;
ð20Þ ð21Þ ð22Þ
the unified description of quantum gravitation and electromagnetism is now proposed as ðrGr;EM mÞWGr;EM ¼ 0:
ð23Þ
Universal applicability of physical law for different observers (‘‘relativity’’) is governed by the invariant hypernumber modulus theorem. Applied to jrGr;EM WGr;EM j ¼ jmWGr;EM j ¼ jmjjWGr;EM j ¼ jrGr;EM jjWGr;EM j;
ð24Þ
it demands identical values for jmj, jWGr,EMj, and j$Gr,EMj in equivalent frames of reference, under the influence of any force. 4. Mixing gravity and electromagnetism The transformation WGr;EM ! WGr;EM ei0 v leaves the modulus jWGr,EMj unchanged and introduces a physical force field Al which acts on the particle’s electric charge e. Depending on the mixing angle a, the effect of this force is either traditional electromagnetism (a = p/2), proposed quantum gravity (a = 0), or a combination thereof. Since a change in a generally changes jWGr,EMj, a must remain constant under space-time coordinate transformation. Therefore, there are three constant properties of a spin 1/2 particle in this formulation which characterize its behavior under the influence of a force: m, e, and a. With a general line element (as in [3], relations 4 through 7) dsQ1 :¼ ð0; dx0 ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0Þ; dsQ2 :¼ ð0; 0; 0; 0; 0; dx3 ; dx2 ; dx1 ; 0; 0; 0; 0; 0; 0; 0; 0Þ;
ð25Þ ð26Þ
dscon16 :¼ dsQ1 þ exp ðai0 ÞdsQ2 ; dT :¼ jdscon16 j ¼ jdsQ1 þ exp ðai0 ÞdsQ2 j;
ð27Þ ð28Þ
one can now examine the space-time transformation P behaviorPof a free particle of general a. The conic sedenion modulus jzj of a general sedenion z ¼ a þ 7n¼0 bn in þ 7n¼0 cn en þ di0 is [7] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2ffi u 7 7 7 u X X X 4 b2n c2n d 2 þ 4 ad bn c n ð29Þ jzj ¼ t a2 þ n¼0
n¼0
n¼0
2
and using j d~ xj :¼ dx21 þ dx22 þ dx23 yields r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 dx20 þ cos2 ajd~ xj2 sin2 ajd~ xj2 þ 4 cos a sin ajd~ xj2 jdscon16 j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4 ¼ dx40 þ jd~ xj4 þ 2dx20 jd~ xj2 cos2 a sin2 a :
ð30Þ
This relation cannot be represented anymore P in a second order tensor formalism (e.g. through the metric tensor glm from General Relativity with ds2 ¼ glm dxl dxm ), but contains mixing effects between quantum gravity and electromagnetism that cannot be separated into their constituent forces. It is also distinct from a potential P fourth order tensor formalism ds4 ¼ glmqr dxl dxm dxq dxr due to the non-associativity of conic sedenions.
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J. Ko¨plinger / Applied Mathematics and Computation 188 (2007) 948–953
For all known particles from the Standard Model (SM) in physics, the relative strength of the gravitational force is docents of orders of magnitude weaker compared to the electrical force, and writing a :¼ p2 b allows to approximate b2 in the order of the relative force strength b2 FGr/FEM m2/e2 (FGr and FEM are the static gravitational and electromagnetic forces between two particles with mass m and electrical charge e; e.g. for two electrons b2e 1042 ). Using c :¼ ð1 jd~ xj2 =dx20 Þ1=2 yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 2 xj þ 2dx20 jd~ xj ðb2 1Þ ð31Þ jdscon16 jðSMÞ dx40 þ jd~ ! dx20 jd~ xj2 2 xj Þ 1 þ b2 ðdx20 jd~ ð32Þ 2 ðdx20 jd~ xj2 Þ 2
2
2
xj þ ðbcÞ jd~ xj : ¼ dx20 jd~
ð33Þ P
This relation could be expressed in second order tensor form ds2 ¼ glm dxl dxm as currently used in physics. The factor (bc)2 depends on the speed of an observer, which is the result of the two forces gravity and electromagnetism being proposed on different space-time geometries (see in detail [3]). Energy levels for which (33) would exhibit a significant deviation from traditional physics are currently only presumed shortly after the Big Bang. For comparison, the Tevatron accelerator (Fermilab) reaches c = Ekin/E0 103 with protons (b2p 1035 ), and LEP (CERN) reached c 2 · 105 for electrons (b2e 1042 ). In both cases there is 2 (bc)2 [ 1029 which makes the traditional length element of a free particle ds2SRT ¼ dx20 j d~ xj an approximation that is indistinguishable from conic sedenion formulation in these experiments. 5. Conclusion and outlook Conic sedenion arithmetic has been used to describe both the classical hyperbolic Dirac equation with electromagnetic field and a counterpart on circular geometry proposed for quantum gravity. The same field Al from electromagnetism is also generator of gravitational interaction in this description. The observed difference between the two forces is quantified through a mixing angle a, which becomes a third particle property next to its electrical charge e and mass at rest m. At experimentally accessible energy levels, this formalism approximates to customary second-order tensor form and has a in the order of p2 a m=e. While effects from conic sedenion formulation are eluding current experimental validation, one is able to describe these two forces on the quantum level in a narrow, closed arithmetic, without unsparing use of extra dimensions, fields, or free parameters. Hypothetical particle types which exhibit predominantly gravitational coupling (a < p/4) could enter into strongly bound states that cannot be broken in current experiments, making its constituents not directly observable. Conic sedenion arithmetic allows to vary the parameter a and therefore is offered for exploration of such states and observables that may indirectly materialize therefrom. Split-octonions, which are computationally equivalent to hyperbolic octonions, have been shown to describe classical electromagnetism on a non-traditional number system [8]. Description of gravity on an Euclidean background has been called ‘‘. . . the only sane way to do quantum gravity nonperturbatively’’ [9], and a genuine Euclidean metric (without ‘‘Wick rotated’’ time element dt ! i dt) was included in the approach. For both these recent developments, conic sedenions with their circular (Euclidean) and hyperbolic (Minkowski) subalgebras may pose a valid computational tool for further examination, in addition to ‘‘asymptotically anti-de Sitter space’’ suggested by Hawking. Extension of the invariant hypernumber modulus theorem to other hypernumber types offers mathematical versatility beyond gravitation and electromagnetism. For example, Muse`s mentioned about w arithmetic that ‘‘at least 3 orders of w (w1,2,3) exist [ ]’’ which form an anti-commutative 3 cycle [10], similar to a typical representation of SU(2) symmetry currently used for the weak interaction in physics. Another example is the power orbit ea which does not fall together with the exponential orbit expae of the same base number e. Current description in physics uses exponential orbits for all forces of the Standard Model, whereas power orbits would be the natural transformation with respect to the invariant hypernumber modulus theorem: W ! Web (b real) leaves jWj unchanged, and so do compound constructs like W ! Wia0 eb wc or W ! Wiaebwc (a, b, c real). Further investigation into these and similar relations is needed in order to conclusively evaluate hypernumbers for the description of physical law.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
J. Ko¨plinger, Dirac equation on hyperbolic octonions, Appl. Math. Comput. 182 (2006) 443–446. J. Ko¨plinger, Signature of gravity in conic sedenions, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.10.049. J. Ko¨plinger, Hypernumbers and relativity, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.10.051. V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Quantum Electrodynamics, second ed., Pergamon Press, 1982. K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions, Appl. Math. Comput. 28 (1988) 47–72. C. Muse`s, Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries, Appl. Math. Comput. 6 (1980) 63–94. K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions – further results, Appl. Math. Comput. 84 (1997) 27–47. M. Gogberashvili, Octonionic electrodynamics, J. Phys. A: Math. Gen. 39 (2006) 7099–7104. S.W. Hawking, Information loss in black holes, Phys. Rev. D 72 (2005) 084013. C. Muse`s, Computing in the bio-sciences with hypernumbers: a survey, Int. J. Bio-Med. Comput. 10 (1979) 519–525.
Gravity and electromagnetism on conic sedenions Jens Ko¨plinger 406 Aberdeen Ter, Greensboro, NC 27403, USA
Abstract Conic sedenions from C. Muse`s’ hypernumbers program are able to express the Dirac equation in physics through their hyperbolic subalgebra, together with a counterpart on circular geometry that has earlier been proposed for description of gravity. An electromagnetic field will now be added to this formulation and shown to be equivalent to current description in physics. With use of an invariant hypernumber modulus condition, a description of quantum gravity with field will be derived. The resulting geometry reduces in very good approximation to relations expressible through customary tensor algebra. However, deviations are apparent at extreme energy levels, as shortly after the Big Bang, that require genuine conic sedenion arithmetic for their correct description. This is offered as method for exploration into bound quantum states, which are not directly observable in the experiment at this time. Extendibility of the invariant modulus condition to higher hypernumber levels promises mathematical flexibility beyond gravity and electromagnetism. 2006 Elsevier Inc. All rights reserved. Keywords: Hypernumbers; Quantum gravity; Electromagnetism, Conic complex numbers; Countercomplex numbers; Sedenions; Octonions; Physics on hyperbolic and circular geometry
1. Introduction In order to qualify and sufficiently support a quantum theory of gravitation on genuine conic sedenion arithmetic (from C. Muse`s’ hypernumbers program), it was first shown that the Dirac equation in physics is expressible on hyperbolic octonion arithmetic [1]. Rotation in the (1, i0) plane yields a counterpart on circular (Euclidean) geometry, which exhibits certain primitive properties of quantum gravity [2]. For large-body (non-quantum) physics, an alignment program was developed from an invariant modulus condition [3], and shown to be consistent with the General Relativity formalism for gravity. This paper will conclude definition of the computational framework for a proposed quantum theory of gravitation and electromagnetism on hypernumbers, by supplying a physical force field to the formulation. An electromagnetic field will be added to the hyperbolic subalgebra and shown to be equivalent to current description in physics, supporting a conjecture by analogy for the circular subalgebra to describe quantum gravitational interaction.
E-mail address: [email protected] 0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.10.050
J. Ko¨plinger / Applied Mathematics and Computation 188 (2007) 948–953
949
Certain mixing effects become apparent at extreme energies, which cannot be separated into the individual constituent forces of gravity and electromagnetism anymore. This makes conic sedenion arithmetic a needed tool for further investigation into such effects. Referring to the power orbit concept, it will be remarked that hypernumbers offer additional mathematical versatility to satisfy a generalized invariant hypernumber modulus theorem, beyond the description of gravity and electromagnetism herein. 2. Electromagnetism The Dirac equation with electromagnetic field is a fundamental building block in describing dynamic interaction of spin 1/2 particles (like electrons or protons). In a simple form it can be written as ½cð^ p eAÞ mW ¼ 0;
ð1Þ
(from [4] equation 32.1) and contains an operator p^ and certain implicit summations. In order to map this relation to conic sedenion arithmetic as in [1], it will now be written explicitely1 in terms of space-time derivatives ol. The terms c, ^ p, A, and W in (1) are summed over three indices l, m, r 2 {0, 1, 2, 3}, with implicit use of a metric tensor glm (describing Minkowski metric; glm = 0 for l5m, g00 = 1, and gii = 1 otherwise), and result in four equations now indexed with q 2 {0, 1, 2, 3}. The c are constant, anti-commuting 4 · 4 matrices [cl]qr on complex numbers, ^ p are the quantum mechanical operators for energy ^p0 io0 and momentum ^pi ioi (i 2 {1, 2, 3}), e is the electric charge of the spin 1/2 particle, m is its mass, and Am the electromagnetic field actingPupon it. Both Am and Wr are a function of time t x0 and space ~ x fx1 ; x2 ; x3 g. With Al :¼ 3m¼0 glm Am one can then write (1) in the following form: ! 3 3 X X ½cl qr glm ð^pm eAm Þ dqr m wr ¼ 0; ð2Þ r¼0
l;m¼0
3 3 X X r¼0
i½cl qr ol
½cl qr eAl
! dqr m wr ¼ 0:
ð3Þ
l¼0
Interpreting the summation terms i½cl qr ol ½cl qr eAl as coefficients ol and eAl to bases icl and cl respectively, one finds the following algebraic relations: (1) All 8 bases icl and cl, as well as, the term m (to basis ‘‘1’’ dqr) are linear independent. (2) Since all cl anti-commute (clcm = cmcl for l5m), the bases to the ol factors anti-commute, as well as the bases to the eAl . (3) The eAl bases anti-commute with bases to om factors for l5m, and commute for l = m. Conic sedenion arithmetic to basis elements bcon16 2 {1, i1, . . . , i7, i0, e1, . . . , e7} and the definitions rEM con16 :¼ ð0; o0 ; 0; 0; 0; eA3 ; eA2 ; eA1 ; 0; eA0 ; 0; 0; 0; o3 ; o2 ; o1 Þ; r i r i r i r i WEM con16 :¼ w0 ; w0 ; w1 ; w1 ; 0; 0; 0; 0; 0; 0; 0; 0; w2 ; w2 ; w3 ; w3 ; with m (m, 0, . . . , 0) allow to express (3) as EM rcon16 m WEM con16 ¼ 0:
1
ð4Þ ð5Þ
ð6Þ
Please note that physical constants c, h, and G are set to 1, and all summations will be written using lower indices only. A metric tensor will be specified if present.
950
J. Ko¨plinger / Applied Mathematics and Computation 188 (2007) 948–953
Proof. For Al = 0 relation (6) reduces to the correct form of the Dirac equation without field on hyperbolic octonion algebra.2 Embedded in conic sedenions, demanding invariance under a transformation of the wave function EM i0 v WEM con16 ! Wcon16 e ;
ð7Þ
in direct analogy to its description in physics (e.g. [4] relation 32.3) with ð8Þ
ol v :¼ eAl ; yields i1 o0 ! i1 ðo0 i0 eA0 Þ ¼ i1 o0 þ e1 eA0 ;
ð9Þ
e5 o3 ! e5 ðo3 þ i0 eA3 Þ ¼ e5 o3 þ i5 eA3 ; e6 o2 ! e6 ðo2 i0 eA2 Þ ¼ e6 o2 i6 eA2 ;
ð10Þ ð11Þ
e7 o1 ! e7 ðo1 þ i0 eA1 Þ ¼ e7 o1 þ i7 eA1 :
ð12Þ
The number bases of the Al terms correspond to (4), and the conic sedenion bases {i1, e5, e6, e7} and {e1, i5, i6, i7} associated with the ol and Al terms respectively satisfy above algebraic relations of the icl and cl bases used in physics (change of sign in Al ¼ Al does not alter these relations). This concludes the proof. h 3. Gravity In an identical procedure to electromagnetism above, the circular octonionic Dirac equation proposed for the description of gravity in [2] (relation 9 there) can be extended by a field eA0 using the definitions rGr con16 :¼ ð0; o0 ; 0; 0; 0; o3 ; o2 ; o1 ; 0; eA0 ; 0; 0; 0; eA3 ; eA2 ; eA1 Þ; WGr con16
:¼
ðwr0 ; wi0 ; wr1 ; wi1 ; wr2 ; wi2 ; wr3 ; wi3 ;
0; 0; 0; 0; 0; 0; 0; 0Þ;
ð13Þ ð14Þ
to Gr ðrGr con16 mÞWcon16 ¼ 0:
This is obtained by demanding invariance of WGr con16
!
i0 v WGr con16 e ;
ð15Þ WGr con16
under the same transformation as (7) above, ð16Þ
(with identical definition of olv := eAl using electromagnetic field Al and electric charge e of the particle). Such an ansatz implies that every spin 1/2 particle with mass at rest m also has an associated electrical charge e. While this is true for most fundamental particles (electron, muon, tau, all quarks, and associated anti-particles), neutrinos are currently assumed to have a mass at rest but no electric charge. Since they only interact through gravitation and the weak force (with associated weak charge), a formalism that would describe both gravity and weak interaction could be more appropriate for these particles. They are therefore excluded from the current investigation. In order to define the computational framework for a proposed quantum theory of gravitation and electromagnetism on hypernumbers, the invariant hypernumber modulus theorem from [3] will now be proposed to be valid for quantum gravity. This theorem was concluded to be sufficiently founded for large-body (nonquantum) gravitation through the NatAliE equations program, and rotation in the (1, i0) plane appears to be a plausible way to transform proven relations for electromagnetism on hyperbolic geometry into gravity on circular geometry.
2
See [1] Eq. (5), but note that definition (4) there is incorrect and should be $hyp8 := (m, o0, 0, 0, 0, o3, o2, o1), as well as definition (3) should be Whyp8 :¼ ðwr0 ; wi0 ; wr1 ; wi1 ; wr2 ; wi2 ; wr3 ; wi3 Þ. The different definitions were the result of using a conic sedenion multiplication table which identified the classical octonion element ‘‘l’’ with sedenion element i4 instead of i4, therefore not being consistent with the cited sources [5,6].
J. Ko¨plinger / Applied Mathematics and Computation 188 (2007) 948–953
951
With use of a real-number mixing angle a and the definitions :¼ ð0; o0 ; 0; 0; 0; 0; 0; 0; 0; eA0 ; 0; 0; 0; 0; 0; 0Þ; rGr;EM Q1
ð17Þ
rGr;EM Q2
ð18Þ
r
Gr;EM
WGr;EM Q1 WGr;EM Q2
:¼ ð0; 0; 0; 0; 0; o3 ; o2 ; o1 ; 0; 0; 0; 0; 0; eA3 ; eA2 ; eA1 Þ; :¼
rGr;EM Q1
:¼
ðwr0 ; wi0 ; wr1 ; wi1 ; 0; 0; 0; 0;
:¼
ð0; 0; 0; 0; wr2 ; wi2 ; wr3 ; wi3 ; WGr;EM þ expði0 aÞWGr;EM ; Q1 Q2
WGr;EM :¼
þ
expði0 aÞrGr;EM ; Q2
ð19Þ
0; 0; 0; 0; 0; 0; 0; 0Þ; 0; 0; 0; 0; 0; 0; 0; 0Þ;
ð20Þ ð21Þ ð22Þ
the unified description of quantum gravitation and electromagnetism is now proposed as ðrGr;EM mÞWGr;EM ¼ 0:
ð23Þ
Universal applicability of physical law for different observers (‘‘relativity’’) is governed by the invariant hypernumber modulus theorem. Applied to jrGr;EM WGr;EM j ¼ jmWGr;EM j ¼ jmjjWGr;EM j ¼ jrGr;EM jjWGr;EM j;
ð24Þ
it demands identical values for jmj, jWGr,EMj, and j$Gr,EMj in equivalent frames of reference, under the influence of any force. 4. Mixing gravity and electromagnetism The transformation WGr;EM ! WGr;EM ei0 v leaves the modulus jWGr,EMj unchanged and introduces a physical force field Al which acts on the particle’s electric charge e. Depending on the mixing angle a, the effect of this force is either traditional electromagnetism (a = p/2), proposed quantum gravity (a = 0), or a combination thereof. Since a change in a generally changes jWGr,EMj, a must remain constant under space-time coordinate transformation. Therefore, there are three constant properties of a spin 1/2 particle in this formulation which characterize its behavior under the influence of a force: m, e, and a. With a general line element (as in [3], relations 4 through 7) dsQ1 :¼ ð0; dx0 ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0Þ; dsQ2 :¼ ð0; 0; 0; 0; 0; dx3 ; dx2 ; dx1 ; 0; 0; 0; 0; 0; 0; 0; 0Þ;
ð25Þ ð26Þ
dscon16 :¼ dsQ1 þ exp ðai0 ÞdsQ2 ; dT :¼ jdscon16 j ¼ jdsQ1 þ exp ðai0 ÞdsQ2 j;
ð27Þ ð28Þ
one can now examine the space-time transformation P behaviorPof a free particle of general a. The conic sedenion modulus jzj of a general sedenion z ¼ a þ 7n¼0 bn in þ 7n¼0 cn en þ di0 is [7] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 !2ffi u 7 7 7 u X X X 4 b2n c2n d 2 þ 4 ad bn c n ð29Þ jzj ¼ t a2 þ n¼0
n¼0
n¼0
2
and using j d~ xj :¼ dx21 þ dx22 þ dx23 yields r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 4 dx20 þ cos2 ajd~ xj2 sin2 ajd~ xj2 þ 4 cos a sin ajd~ xj2 jdscon16 j ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4 ¼ dx40 þ jd~ xj4 þ 2dx20 jd~ xj2 cos2 a sin2 a :
ð30Þ
This relation cannot be represented anymore P in a second order tensor formalism (e.g. through the metric tensor glm from General Relativity with ds2 ¼ glm dxl dxm ), but contains mixing effects between quantum gravity and electromagnetism that cannot be separated into their constituent forces. It is also distinct from a potential P fourth order tensor formalism ds4 ¼ glmqr dxl dxm dxq dxr due to the non-associativity of conic sedenions.
952
J. Ko¨plinger / Applied Mathematics and Computation 188 (2007) 948–953
For all known particles from the Standard Model (SM) in physics, the relative strength of the gravitational force is docents of orders of magnitude weaker compared to the electrical force, and writing a :¼ p2 b allows to approximate b2 in the order of the relative force strength b2 FGr/FEM m2/e2 (FGr and FEM are the static gravitational and electromagnetic forces between two particles with mass m and electrical charge e; e.g. for two electrons b2e 1042 ). Using c :¼ ð1 jd~ xj2 =dx20 Þ1=2 yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 2 xj þ 2dx20 jd~ xj ðb2 1Þ ð31Þ jdscon16 jðSMÞ dx40 þ jd~ ! dx20 jd~ xj2 2 xj Þ 1 þ b2 ðdx20 jd~ ð32Þ 2 ðdx20 jd~ xj2 Þ 2
2
2
xj þ ðbcÞ jd~ xj : ¼ dx20 jd~
ð33Þ P
This relation could be expressed in second order tensor form ds2 ¼ glm dxl dxm as currently used in physics. The factor (bc)2 depends on the speed of an observer, which is the result of the two forces gravity and electromagnetism being proposed on different space-time geometries (see in detail [3]). Energy levels for which (33) would exhibit a significant deviation from traditional physics are currently only presumed shortly after the Big Bang. For comparison, the Tevatron accelerator (Fermilab) reaches c = Ekin/E0 103 with protons (b2p 1035 ), and LEP (CERN) reached c 2 · 105 for electrons (b2e 1042 ). In both cases there is 2 (bc)2 [ 1029 which makes the traditional length element of a free particle ds2SRT ¼ dx20 j d~ xj an approximation that is indistinguishable from conic sedenion formulation in these experiments. 5. Conclusion and outlook Conic sedenion arithmetic has been used to describe both the classical hyperbolic Dirac equation with electromagnetic field and a counterpart on circular geometry proposed for quantum gravity. The same field Al from electromagnetism is also generator of gravitational interaction in this description. The observed difference between the two forces is quantified through a mixing angle a, which becomes a third particle property next to its electrical charge e and mass at rest m. At experimentally accessible energy levels, this formalism approximates to customary second-order tensor form and has a in the order of p2 a m=e. While effects from conic sedenion formulation are eluding current experimental validation, one is able to describe these two forces on the quantum level in a narrow, closed arithmetic, without unsparing use of extra dimensions, fields, or free parameters. Hypothetical particle types which exhibit predominantly gravitational coupling (a < p/4) could enter into strongly bound states that cannot be broken in current experiments, making its constituents not directly observable. Conic sedenion arithmetic allows to vary the parameter a and therefore is offered for exploration of such states and observables that may indirectly materialize therefrom. Split-octonions, which are computationally equivalent to hyperbolic octonions, have been shown to describe classical electromagnetism on a non-traditional number system [8]. Description of gravity on an Euclidean background has been called ‘‘. . . the only sane way to do quantum gravity nonperturbatively’’ [9], and a genuine Euclidean metric (without ‘‘Wick rotated’’ time element dt ! i dt) was included in the approach. For both these recent developments, conic sedenions with their circular (Euclidean) and hyperbolic (Minkowski) subalgebras may pose a valid computational tool for further examination, in addition to ‘‘asymptotically anti-de Sitter space’’ suggested by Hawking. Extension of the invariant hypernumber modulus theorem to other hypernumber types offers mathematical versatility beyond gravitation and electromagnetism. For example, Muse`s mentioned about w arithmetic that ‘‘at least 3 orders of w (w1,2,3) exist [ ]’’ which form an anti-commutative 3 cycle [10], similar to a typical representation of SU(2) symmetry currently used for the weak interaction in physics. Another example is the power orbit ea which does not fall together with the exponential orbit expae of the same base number e. Current description in physics uses exponential orbits for all forces of the Standard Model, whereas power orbits would be the natural transformation with respect to the invariant hypernumber modulus theorem: W ! Web (b real) leaves jWj unchanged, and so do compound constructs like W ! Wia0 eb wc or W ! Wiaebwc (a, b, c real). Further investigation into these and similar relations is needed in order to conclusively evaluate hypernumbers for the description of physical law.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
J. Ko¨plinger, Dirac equation on hyperbolic octonions, Appl. Math. Comput. 182 (2006) 443–446. J. Ko¨plinger, Signature of gravity in conic sedenions, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.10.049. J. Ko¨plinger, Hypernumbers and relativity, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.10.051. V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Quantum Electrodynamics, second ed., Pergamon Press, 1982. K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions, Appl. Math. Comput. 28 (1988) 47–72. C. Muse`s, Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries, Appl. Math. Comput. 6 (1980) 63–94. K. Carmody, Circular and hyperbolic quaternions, octonions, and sedenions – further results, Appl. Math. Comput. 84 (1997) 27–47. M. Gogberashvili, Octonionic electrodynamics, J. Phys. A: Math. Gen. 39 (2006) 7099–7104. S.W. Hawking, Information loss in black holes, Phys. Rev. D 72 (2005) 084013. C. Muse`s, Computing in the bio-sciences with hypernumbers: a survey, Int. J. Bio-Med. Comput. 10 (1979) 519–525.