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On the nature of complex time, diffusion and the two-slit experiment
Chaos
Solitom
& Fmctalr
Vol. 5, No. 6, pp. 1031-1032, 1995 Elsevier Science, Ltd Printed in Great Britain 0960~0779/95$9.50 + .cm
On the Nature of Complex Time, Diffusion and the Two-slit Experiment M. S. EL NASCXIE DAh4TP, Cambridge, UK
Motivated by the negative sign of the time component in the metric of the theory of relativity, the concept of imaginary time transformation (t + it; i = d-1) was applied by Hawking in a way which enabled him to smear out the singularity of Einstein’s equations and convert it to practically a simple starting point on a boundaryless space-time manifold [l]. In the present short note we would like to argue that it could be interpreted as a complex time 0 + it and since complex numbers appear only in pairs, the conjugate complex time would be 0 - it. This idea could then be developed in two different but related directions. First, using it, the Schriidinger equation could be transformed to a classical diffusion equation, while -it would transform the same equation to the conjugate i.e. dual diffusion equation. In other words, the set of Schriidinger equations describing retarded and advanced wave functions are replaced in that way by a set of forward and backward running diffusion equations [2]. It is interesting to note that achieving this result was the goal which SchrGdinger set for himself in 1931 although without reaching a final conclusion as he had hoped for [3], because he could not account for interference in classical diffusion processes[2,4]. Our second remark is with regard to the sharp distinction which is experienced in the ‘real’ world between past, present and future. Interpreting -it within the theory of Cantorian space-time [4] as a one-dimensional time past set while +it is the set of time to come, then the present or nowness could be seen as the intersection set between the = t2 and ‘advanced’ and the ‘retarded’ sets. That way we come to write T = (-it)(+it) consequently the one-dimensional ‘nowness’ time which is the only ‘real’ time would be t = J/T. Here the time t2 may be related to the idea of dual time scales advanced by J. Haldane [S]. We hasten here to say that such a combination of time past and time to come is by no means entirely new since this is basically what is done every time we calculate probabilities using Born interpretation of the state vector or wave function of the Schriidinger equation namely+ P = I+!J)(~a)). Th’1sis the case because 1~) is the solution of the retarded Schriidinger equation while I&) is the solution of the advanced conjugate Schriidinger wave travelling back in time [2]. This is obviously related to what is known in cosmology as the Eddington-Lemaitre-Bondi universe. The present interpretation, simple as it may be, could nevertheless have considerable implications for the way the whole of quantum mechanics could be interpreted in a ‘realistic’ way. Travelling back in time implies velocities exceeding c which is thought superficially to be forbidden by the special theory of relativity. Yet given the right +Using Dirac’s ket and bra notation this could be written (more elegantly) as ($1 q) = P. 1031
1032
M. S. EL NASCHIE!
interpretation it is easily shown that special relativity imposes such a restriction only on accelerating a body or a wave beyond the speed of light (c) and does not in any a priori manner preclude the existence of an instantaneous correlation as is most natural in a Cantorian space-time [4] and as clearly implied by the outcome of many real tests inspired by the EPR Gedanken experiment [6,7]. Note here that in its most general form no distinction is made whatsoever between time and space within the Cantorian theory of space-time and unlike the theory of relativity. Strictly speaking Cantorian space-time is a multidimensional Cantorian space rather than space-time [4]. In addition it is obvious that from the point of view of the masslessphotons, time does not flow at all. Furthermore, the possibility of massless superluminal particles like phonons and excitons are in the mean time accepted hypotheses which may play a profound role in explaining, for instance, superconductivity. Finally, even more recently than that a proposition was made towards re-interpreting the old idea of ether and the concept of Cantorian (fractal) space-time in terms of massless and energyless informational field with the hypothetical informion as its quanta [2]. Such a particle could, of course, not be detectable under any classical physical experimental conditions directly. However, the presence of such particles may be inferred indirectly by the way they influence potential geodesics and determine the geometrical distribution of matter particles such as electrons and energy particles such as photons in the formation of patterns. A very possible candidate for the indirect manifestation of such particles may very well be the appearance and disappearance of interference patterns in the two-slit experiment particularly in the form proposed by Scully, Walther and their associates [4,8]. In conclusion we may point out the conceptual similarity between the DNA-like informational ether or Cantorian micro space and the MacKay naked gene and the quasicrystal matrix on which it is possible to encode information as if one would be writing a message on an abacus [9]. In a forthcoming paper we will show using the BanachTarski theorem how Cantorian space-time itself could have been created from the initial singularity [lo]. REFERENCES 1. S. W. Hawking, Nucl. Phys. B 239, 257 (1984). 2. M. S. El Naschie, A note on quantum mechanics, difusional interference and informions, Chaos, Solitons & Fructuzs, 5(S), 881-884 (1995). 3. E. Schradinger, Ueber die Umkehrung der Naturgesetxe. Sitzungsberichte der preussischen Akad. der Wissenschaften, physikalische mathematische klasse, pp. 144-153 (1931). 4. M. S. El Naschie, 0. E. Rossler and Y. Prigogine, Quantum Mechanics, Diffusion and Chaotic Fractals. Elsevier Science, Oxford (1995). 5. J. S. Haldane, Nature 139, 1092 (1937) and 158, 555 (1944). 6. A. Aspect, J. Dahbard and G. Roger, Experimental test of Bell’s inequality using time varying analyzers. Phys. Rev. Lett. 49, 1804-1807 (1982). 7. S. Prasad, M. 0. Scully and W. Martienssen, A quantum description of the beam splitter, Opt. Comm. 62, 139-145 (1987). 8. M. 0. Scully, B. G. Engler and H. Walther, Quantum optical test of complementarity, Nature 351(6322), 111-116 (1991). 9. A. MacKay, The crystal abacus, in Clay Minerals and the Origin of Life, edited by A. Cairns-Smith and H. Hartman, pp. 140-143. Cambridge University Press, Cambridge, UK (1986). 10. M. S. El Naschie, Banach-Tarski theorem and Cantorian micro space-time, Chaos, Solitons & Fractals to appear.
Solitom
& Fmctalr
Vol. 5, No. 6, pp. 1031-1032, 1995 Elsevier Science, Ltd Printed in Great Britain 0960~0779/95$9.50 + .cm
On the Nature of Complex Time, Diffusion and the Two-slit Experiment M. S. EL NASCXIE DAh4TP, Cambridge, UK
Motivated by the negative sign of the time component in the metric of the theory of relativity, the concept of imaginary time transformation (t + it; i = d-1) was applied by Hawking in a way which enabled him to smear out the singularity of Einstein’s equations and convert it to practically a simple starting point on a boundaryless space-time manifold [l]. In the present short note we would like to argue that it could be interpreted as a complex time 0 + it and since complex numbers appear only in pairs, the conjugate complex time would be 0 - it. This idea could then be developed in two different but related directions. First, using it, the Schriidinger equation could be transformed to a classical diffusion equation, while -it would transform the same equation to the conjugate i.e. dual diffusion equation. In other words, the set of Schriidinger equations describing retarded and advanced wave functions are replaced in that way by a set of forward and backward running diffusion equations [2]. It is interesting to note that achieving this result was the goal which SchrGdinger set for himself in 1931 although without reaching a final conclusion as he had hoped for [3], because he could not account for interference in classical diffusion processes[2,4]. Our second remark is with regard to the sharp distinction which is experienced in the ‘real’ world between past, present and future. Interpreting -it within the theory of Cantorian space-time [4] as a one-dimensional time past set while +it is the set of time to come, then the present or nowness could be seen as the intersection set between the = t2 and ‘advanced’ and the ‘retarded’ sets. That way we come to write T = (-it)(+it) consequently the one-dimensional ‘nowness’ time which is the only ‘real’ time would be t = J/T. Here the time t2 may be related to the idea of dual time scales advanced by J. Haldane [S]. We hasten here to say that such a combination of time past and time to come is by no means entirely new since this is basically what is done every time we calculate probabilities using Born interpretation of the state vector or wave function of the Schriidinger equation namely+ P = I+!J)(~a)). Th’1sis the case because 1~) is the solution of the retarded Schriidinger equation while I&) is the solution of the advanced conjugate Schriidinger wave travelling back in time [2]. This is obviously related to what is known in cosmology as the Eddington-Lemaitre-Bondi universe. The present interpretation, simple as it may be, could nevertheless have considerable implications for the way the whole of quantum mechanics could be interpreted in a ‘realistic’ way. Travelling back in time implies velocities exceeding c which is thought superficially to be forbidden by the special theory of relativity. Yet given the right +Using Dirac’s ket and bra notation this could be written (more elegantly) as ($1 q) = P. 1031
1032
M. S. EL NASCHIE!
interpretation it is easily shown that special relativity imposes such a restriction only on accelerating a body or a wave beyond the speed of light (c) and does not in any a priori manner preclude the existence of an instantaneous correlation as is most natural in a Cantorian space-time [4] and as clearly implied by the outcome of many real tests inspired by the EPR Gedanken experiment [6,7]. Note here that in its most general form no distinction is made whatsoever between time and space within the Cantorian theory of space-time and unlike the theory of relativity. Strictly speaking Cantorian space-time is a multidimensional Cantorian space rather than space-time [4]. In addition it is obvious that from the point of view of the masslessphotons, time does not flow at all. Furthermore, the possibility of massless superluminal particles like phonons and excitons are in the mean time accepted hypotheses which may play a profound role in explaining, for instance, superconductivity. Finally, even more recently than that a proposition was made towards re-interpreting the old idea of ether and the concept of Cantorian (fractal) space-time in terms of massless and energyless informational field with the hypothetical informion as its quanta [2]. Such a particle could, of course, not be detectable under any classical physical experimental conditions directly. However, the presence of such particles may be inferred indirectly by the way they influence potential geodesics and determine the geometrical distribution of matter particles such as electrons and energy particles such as photons in the formation of patterns. A very possible candidate for the indirect manifestation of such particles may very well be the appearance and disappearance of interference patterns in the two-slit experiment particularly in the form proposed by Scully, Walther and their associates [4,8]. In conclusion we may point out the conceptual similarity between the DNA-like informational ether or Cantorian micro space and the MacKay naked gene and the quasicrystal matrix on which it is possible to encode information as if one would be writing a message on an abacus [9]. In a forthcoming paper we will show using the BanachTarski theorem how Cantorian space-time itself could have been created from the initial singularity [lo]. REFERENCES 1. S. W. Hawking, Nucl. Phys. B 239, 257 (1984). 2. M. S. El Naschie, A note on quantum mechanics, difusional interference and informions, Chaos, Solitons & Fructuzs, 5(S), 881-884 (1995). 3. E. Schradinger, Ueber die Umkehrung der Naturgesetxe. Sitzungsberichte der preussischen Akad. der Wissenschaften, physikalische mathematische klasse, pp. 144-153 (1931). 4. M. S. El Naschie, 0. E. Rossler and Y. Prigogine, Quantum Mechanics, Diffusion and Chaotic Fractals. Elsevier Science, Oxford (1995). 5. J. S. Haldane, Nature 139, 1092 (1937) and 158, 555 (1944). 6. A. Aspect, J. Dahbard and G. Roger, Experimental test of Bell’s inequality using time varying analyzers. Phys. Rev. Lett. 49, 1804-1807 (1982). 7. S. Prasad, M. 0. Scully and W. Martienssen, A quantum description of the beam splitter, Opt. Comm. 62, 139-145 (1987). 8. M. 0. Scully, B. G. Engler and H. Walther, Quantum optical test of complementarity, Nature 351(6322), 111-116 (1991). 9. A. MacKay, The crystal abacus, in Clay Minerals and the Origin of Life, edited by A. Cairns-Smith and H. Hartman, pp. 140-143. Cambridge University Press, Cambridge, UK (1986). 10. M. S. El Naschie, Banach-Tarski theorem and Cantorian micro space-time, Chaos, Solitons & Fractals to appear.