With the theory of special relativity, time has been linked with space into a
four-dimensional space-time from which a basic question must be asked: can
space be really transformed into time and vice-versa? The response is
affirmative if time has the same structural topological structure as space at
the subplanckian quantum level in such a way that a discrete structural quantum
time constitutes the time part of the space-time internal vacuum of every
elementary particle. It has thus been shown that a quantum time, quantized
algebraically according to a lattice of time quanta, really exists and is
emergent in the sense that time quanta can be transformed into space quanta and
vice-versa. Furthermore,this quantum time, only relevant at the subplanckian
scale, is proved to be in one-to-one correspondence with the absolute and
relative clock times.Comment: 8 page

Pierre, Christian

English

arXiv

With the theory of special relativity, time has been linked with space into a
four-dimensional space-time from which a basic question must be asked: can
space be really transformed into time and vice-versa? The response is
affirmative if time has the same structural topological structure as space at
the subplanckian quantum level in such a way that a discrete structural quantum
time constitutes the time part of the space-time internal vacuum of every
elementary particle. It has thus been shown that a quantum time, quantized
algebraically according to a lattice of time quanta, really exists and is
emergent in the sense that time quanta can be transformed into space quanta and
vice-versa. Furthermore,this quantum time, only relevant at the subplanckian
scale, is proved to be in one-to-one correspondence with the absolute and
relative clock times.
 Comment: 8 page

DIAL UCLouvain

arXiv:physics/0701242v1  [physics.gen-ph]  22 Jan 2007The time at the subplanckian scaleC. PierreInstitut de Mathe´matique pure et applique´eUniversite´ de LouvainChemin du Cyclotron, 2B-1348 Louvain-la-Neuve, Belgiumpierre@math.ucl.ac.beAbstractWith the theory of special relativity [1], time has been linked withspace into a four-dimensional space-time from which a basic questionmust be asked: can space be really transformed into time and viceversa? The response is affirmative if time has the same structuraltopological structure as space [2] at the subplanckian quantum levelin such a way that a discrete structural quantum time constitutes thetime part [3] of the space-time internal vacuum of every elementaryparticle.It has thus been shown that a quantum time, quantized alge-braically according to a lattice of time quanta, really exists and isemergent in the sense that time quanta can be transformed into spacequanta and vice versa. Furthermore, this quantum time, only relevantat the subplanckian scale, is proved to be in one-to-one correspondencewith the absolute and relative clock times.11 Universal and relative clock timesUnderstanding the nature of time has long been a main challenge for scientists[4]–[8] and philosophers [9]. Indeed, if, at first sight, it is relatively easy torealize that time has a continuous structure composed of a succession ofinstants corresponding to points on the line, a deeper insight of it revealsits close relationship with movement. This is perhaps why Newton [10] inhis attempt to isolate time from the physical reality viewed it as a linearuniversal and absolute mathematical object.Since antiquity until Einstein, time was considered as being exterior to thephysical phenomena [11] allowing the measurement of their evolution. Thisabsolute time, conceived as a universal clock time, measured the runningflow of time and has been treated as a unidimensional object within theframework of the Euclidean geometry.The physical reality was not only described by Newton with respect toa universal clock time but also in function of an absolute three-dimensionalspace: this gave him the absolute mathematical frame allowing to work outthe dynamics of moving macroscopic bodies.In this classical dynamics [12], macroscopic objects move in the referenceframe of the external three-dimensional space with respect to the absoluteclock time. Until special relativity, space and time were considered as inde-pendent concepts although mingled with the dynamics they describe.It was only after the contribution of Minkowski, Lorentz, Poincare [1]and principally Einstein [13] to special relativity that a four-dimensionalcontinuous space-time world was introduced in Physics: it is sometimes calledchronogeometry with reference to the clock time and is characterized by aMinkowski pseudo-Euclidean geometry [1] in which the dimension of time isperpendicular to the three dimensions of space. However, space and time arenot completely equivalent in this context since time is not a fourth dimensionof space.A basic concept of special relativity, allowing the description of the eventsby space-time coordinates (x, y, z, t) , is the inertial reference frame repre-sented by a four-dimensional coordinate system moving at constant velocity.The distance between two space-time events is obtained in the Minkowski-Lorentz geometry from their space-time interval which is the same in allinertial frames: this is equivalent to saying that the space-time interval be-tween two events is an invariant of the transformation from an inertial frame1into another given by the Lorentz transformations.These considerations imply the two following drastic changes in our un-derstanding of time:1. the timelike (i.e. real) interval [14] between two space-time events in-troduces the concept of proper time which is zero for the light in thevacuum in the frame of special relativity but different from zero forall other kind of matter, for example the elementary particles having avelocity inferior to the one of light c .2. the absolute time (and space) of Newton [10], is replaced by a relativetime (and space) associated with each reference body and varying withrespect to its velocity and the other considered reference bodies. Inthis respect, let us quote A. Einstein “Every reference body has itsown particular time; unless we are told the reference body to which thestatement of time refers, there is no meaning in a statement of time ofan event”.Let then s212 be this time like, i.e. causal, space-time interval betweentwo events “ 1 ” and “ 2 ” [14]s212 = c2t212 − r212 > 0where the time and space coordinates of the two events are denoted respec-tively by r and t with r2 = x2 + y2+ z2 in Cartesian coordinates (x, y, z) .This concept of interval between two space-time events is the corner-stone of the Einstein-Minkowski-Lorentz-Poincare chronogeometry [1] whichis rigid, i.e. flat, in the frame of special relativity when it is curved in generalrelativity [15]–[17] due to the presence of matter deforming the neighbouringspace-time. The clock time is thus affected by gravitational forces as it canbe measured that a clock on the earth will run more slowly than a clock notsubjected to gravitation.Special and general relativity then tell us that every moving referencebody, submitted to its related dynamics, has its own proper time:a) calculated with respect to every external (moving) reference body towhich the universal cosmic time can correspond;b) deformed by (external) gravitational forces.2A fundamental question is then the following:Are space and time dependent in the frame of special (and general) rela-tivity?The Lorentz transformation ∆t′ = ∆t(1− v2c2) 12shows that the (proper)time ∆t′ of a body moving at velocity “ v ” is shorter than the correspondingtime ∆t in the fixed inertial frame: in other words, a moving clock runs moreslowly than a clock at rest.Similarly, the Lorentz transformation ∆x′ = ∆x“1− v2c2” 12gives the (proper)length ∆x′ of our body moving at velocity v with respect to the correspond-ing length ∆x in a fixed inertial frame: this shows that there is a dilation oflengths in the moving reference frame with respect to its equivalent at rest.So, according to special relativity, a moving body experiences, in its ownreference frame, time contraction and space dilation, with respect to thefactor(1− v2c2) 12.But, special relativity does not tell us that the contracted time has beentransformed into dilated length: and, this seems a priori not possible sincethe clock time does not have the same topological structure as the three-dimensional space.2 Towards a quantum time of structureTo obtain a new breakthrough in this problem, quantum (field) theory [18]must be taken into account, and, especially its connection [19] with specialrelativity.A first remark can be made: the time used in quantum theories refersmostly to the internal dynamics of the considered microscopic object andappears by means of the frequency “ ν ≃ 1tp” inversely proportional to theperiod tp [20]. Now, this periodical time can be related to the astronomicaltime: there is thus a one-to-one correspondence between these two typesof time. For example, the frequency of the cesium atom is given in termsof a number of oscillations corresponding to one period of one second ofastronomical time.A new insight into this question will be reached if the reference frames ofthe special (and general) relativity are considered at the level of elementary3particles. Indeed, the energy of a particle is given in special relativity by:E =m c2√1− v2c2where m is the rest mass of the particle. For low velocity ( v ≪ c ), thedevelopment of E in power of vcis:E ≈ m c2 +m v22where E = m c2 is the rest energy of the particle.This allows to find the famous relation:E2 = p2 c2 +m2 c4between the square of the energy E of the particle and the square of itslinear momentum p , which implies that:a) if the particle is at rest, i.e. with p = 0 , then E = m c2 givesthe amount of energy which should be obtained by its annihilation inphoton(s) from its rest mass.b) E ≃ p c is the amount of energy resulting from the annihilation of itsrest mass m (case m →˜ 0 ) into photons.Furthermore, if it is taken into account that, in quantum (field) theories,the following differential operators:pi −→~i∂∂i( i = x, y, z )E −→ i~∂∂t(and, by analogy, m→ i~ ∂∂t0where t0 is the proper time)correspond respectively to the i -th component of the linear momentum andto the energy of our particle, it does not seem unwise to consider that themass “m ” and the energy “E ” or the linear momentum “ p ” are localizedin two orthogonal spaces respectively of “time” and “space” type allowingthe internal transfer of mass into energy and vice versa.4This is only possible if these two orthogonal spaces have a topological(spatial) structure as it was noticed before.And, thus, we are forced to admit that elementary particles must beprovided with an internal topological structure as it was introduced by theauthor [3]. Mathematics show that this internal topological structure must beof space (-time) type. And, if we wish to connect [22] quantum field theories(QFT) to general relativity, this internal space (-time) structure must beexpanding in order that the cosmological constant [21] of general relativitycould correspond to it: it then would constitute the fundamental structureof the QFT vacuum shared out amongst the considered elementary particles.In this perspective, every elementary particle would then be characterizedby an internal expanding space-time structure constituting its own vacuumfrom which its matter shell could be generated.It was justified in [3] and in [22] that the internal structure of an elemen-tary particle must be:a) composed of three embedded shells ST ⊂ MG ⊂ M : space-time(ST ), middle-ground (MG ) and mass (M ).b) of bilinear type, i.e. composed of the product of a left semiparticlelocalized in the upper half space by a symmetric right (co-)semiparticlelocalized in the lower half space.Remark that the bilinearity was considered (contrary to QFT whichworks in a mathematical linear frame) in order to really take into ac-count the special relativity invariants in QFT.c) really quantized algebraically in such a way that quanta then are ir-reducible algebraic closed subsets characterized by a Galois extensiondegree N .In the framework of the global program of Langlands [23], the time com-ponent of the most internal space-time shell “ST ” of the vacuum of anelementary particle (mostly, fermion) was shown to be a time string field[3] composed of a double tower of packets of symmetric strings behavinglike harmonic oscillators and characterized by increasing integers labellingthe number of quanta in these. The time string field of a massive fermioncan then generate, from singularities on it, the two covering middle-ground(MG ) and mass (M ) string fields in such a way that the latter correspondsto the proper mass structure. The space string field of the “ST ” shell of a5fermion is orthogonal to the corresponding time string field and has the sametype of structure.Time and space string fields are thus structured according to a lattice ofquanta and are emergent each one with respect to the other in the sense thattime quanta can be transformed into space quanta by passing throughoutthe origin and vice versa.The time string field, which is thus a reservoir of space, is adiscrete quantum time of structure having an (anti)inertial effect onthe corresponding quantum space field by slowing down the space-time field.Every string at n quanta of this time string field is characterized by:1. a quantum periodical proper time t0q(n) which can be interpreted as adistance filled by a set of cycles referring to one quantum in this stringlike the wavelength.2. a quantum frequency νq(n) which gives the number of oscillations perquantum in this time string.In order to connect this quantum time of structure to the clock times, twobasic questions will finally be considered. The first question can be stated asfollows: is there a one-to-one correspondence between the discrete quantumtime of structure and the universal clock time? The reply is affirmativebecause to every time quantum of a string at n quanta corresponds:a) a period t0p(n) which gives the clock time needed for a set of cyclesrelated to this quantum.b) a periodical frequency νp(n) =1t0p(n)defined from t0p(n) .The second question consists in knowing if a quantum time of structurecan be defined with respect to all types of clock times considered in thispaper. It is easy to realize that this will be the case if the quantum propertime t0q(n) and the quantum frequency νq(n) are dependent.Now, it was proved by the author [3] in an attempt to solve the conjec-ture of Riemann [24] that the trivial zeros of the zeta function ζ(s) [25]–[26],related to the structure of strings at n quanta (the integer n varying), arein one-to-one correspondence with the equivalent non-trivial zeros of whichimaginary parts γn are approximately equal to the energies, i.e. the frequen-cies of these strings.6It then results that the only relevant time at the subplanckian quantumscale is the quantum structural time of the space-time shell “ST ” of theinternal vacua of elementary particles in such a way that the universal andrelative clock times be reduced to this quantum time at this microscopicscale.References[1] Lorentz, H., Einstein, A., Minkowski, H. The principle of relativity ,Dover, N.Y. (1923).[2] Atiyah, M. Reflections on geometry and physics, Surveys in Diff. Geom.,2, 1–6 (1995).[3] Pierre, C. Algebraic quantum theory , ArXiv math-ph/0404024 (2004).[4] Penrose, R. Structure of space-time in Battelle rencontres, Ed. de Witt,B., Wheeler, J.A. Benjamin (1968).[5] ’t Hooft, G. How does God play dice? (Pre)-determinism at the Planckscale, ArXiv-hep-th/0104219 (2001).[6] Hawking, S.W., Pennrose, R. The nature of space and time, PrincetonUniv. Press (1996).[7] Hawking, S.W. A brief history of time: From Big-Bang to black holes,Bantam books, N.Y. (1988).[8] Jasselette, P. Le temps des physiciens, Rev. Quest. Sci., 154, 461–477(1983).[9] Wang, W. Time in philosophy and in physics: from Kant and Einsteinto Go¨del, Synthese, 102, 215–234 (1995).[10] Newton, I. Philosophiae Naturalis Principia Mathematica, Londini,Jussu Societatis Regiae ac Typis J. Streater (1687).[11] Damour, T. La relativite´ ge´ne´rale in: Qu’est-ce que l’Univers, Universite´de Tous les Savoirs, Vol. 4, dir. Y. Michaud, Ed. Jacob, O. (2000).7[12] Arnold, V., Avez, A. Proble`mes ergodiques de la me´canique classique,Gauthier-Villars (1967).[13] Einstein, A. Zur Elektrodynamik Bewegter Ko¨rper, Ann. Phys., 17,891–921 (1905).[14] Einstein, A. Quatre confe´rences sur la the´orie de la relativite´, Gauthier-Villars (1971).[15] Einstein, A. La the´orie de la relativite´ restreinte et ge´ne´rale – Expose´e´le´mentaire, Gauthier-Villars (1954).[16] Hawking, S.W., Ellis, G.F.R. The large scale structure of space-time,Cambridge Univ. Press (1973).[17] Misner, C., Thorne, K., Wheeler, J.A. Gravitation, Freeman.[18] Weinberg, S. The quantum theory of fields, Vols. I & II , CambridgeUniv. Press (1994).[19] Dirac, P.A.M. Relativity and quantum mechanics, fields and quanta, 3,139–164 (1972).[20] Feynman, R.P., Leighton, R., Sands, M. The Feynman lectures onphysics, Vol. 1 , Addison Wesley (1964).[21] Weinberg, S. Einstein’s mistakes, Physics Today , 31–36 (2005).[22] Pierre, C. A new track for unifying general relativity with quantum fieldtheories, ArXiv org, gr-qc/0510091 (2005).[23] Pierre, C. N -dimensional global correspondences of Langlands, ArXivorg, math.RT/0510348 (2005).[24] Titschmarsch, E.C. The theory of the Riemann zeta function, OxfordUniv. Press (1967).[25] Berry, M., Keating, J. The Riemann zeros and eigenvalue asymptotics,Siam Rev., 41, 236–266 (1999).[26] Katz, N. Sarnark, P. Zeros of zeta functions and symmetry, Bull. Amer.Math. Soc., 36, 1–26 (1999).8

The time at the subplanckian scale

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