A Wave Group for Entanglement, Linking Uncertainties in Time and Space

Antony J. Bourdillon

UHRL, San Jose, USA.

DOI: 10.4236/jmp.2012.33041                

Abstract

A carrier wave in a 5-dimensional wave group is examined for information on electromagnetic waves and on particle probability amplitudes. Simulations by Maxwell’s equations show that the phase and group velocities in electromagnetic waves are equal, both in vacuo and in dielectric media. By contrast, particle probability amplitudes in wave packet motion are more complicated. A dependence of rest mass on relative phase and group velocities is derived by consistency. Occurrences that are simultaneous and connected on wave fronts in the rest frame, appear separated when observed in moving frames. Uncertainties in space and time are linked by the probability amplitude wave group.

Keywords

Entanglement; Phase Velocity; Wave Group; Electromagnetic Waves; Probability Amplitudes; Uncertainty Principle

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	A. Einstein, B. Podolski and N. Rosen, “Can Quantum Mechanical Description of Physical Reality Be Consid- ered Complete?” Physical Reviews, Vol. 47, No. 10, 1935, pp. 777-780. doi:10.1103/PhysRev.47.777
[2]	J. S. Bell, “On the Problem of Hidden Variables in Quan- tum Mechanics,” Reviews of Modern Physics, Vol. 38, No. 3, 1966, pp. 447-452. doi:10.1103/RevModPhys.38.447
[3]	D. Bohm and J. Bub, “A Proposed Solution of the Meas- urement Problem in Quantum Mechanics by a Hidden Variable Theory,” Reviews of Modern Physics, Vol. 38, No. 3, 1966, pp. 453-475. doi:10.1103/RevModPhys.38.453
[4]	J. Longdell, “Quantum Information: Entanglement on Ice,” Nature, Vol. 469, No. 7331, 2011, pp. 475-476. doi:10.1038/469475a
[5]	C. Clausen, I. Usmani, F. Bussieres, N. Sangouard, M. Afzelius, H. de Riedmatten and N. Gisin, “Quantum Stor- age of Photonic Entanglement in a Crystal,” Nature, Vol. 469, No. 7331, 2011, pp. 508-511. doi:10.1038/nature09662
[6]	E. Saglamurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussieres, M. George, R. Ricken, W. Sohler and W. Tittle, “Broadband Waveguide Quantum Memory for Entangled Photons,” Nature, Vol. 469, No. 7331, 2011, pp. 512-515. doi:10.1038/nature09719
[7]	A. Faipour, X. Xing and A. Steinberg, “Weak Measure- ment Amplification of Single Photon Nonlinearity,” Phy- sical Review Letters, Vol. 107, No. 13, 2011, p. 133603.
[8]	I. A. Arbab, “A New Wave Equation of the Electron,” Journal of Modern Physics, Vol. 2, No. 9, 2011, pp. 1012-1016.
[9]	A. J. Bourdillon, “Use of the Track Structure Approach in TEM, Ultramicroscopy, Vol. 83, No. 3-4, 2000, pp. 261- 264.
[10]	R. Audi, et al. (Ed.), “The Cambridge Dictionary of Philosophy,” Cambridge, 1999.
[11]	M. Born, H. Born and A. Einstein, “The Born-Einstein Letters,” Macmillan, London, 1971.
[12]	A. Einstein, “A Stubbornly Persistent Illusion, the Essen- tial Scientific Works of Albert Einstein,” S. Hawking (Ed.), Running Press, New York, 2007.
[13]	A. Pais, “Subtle Is the Lord, the Science and the Life of Albert Einstein,” Oxford University Press, Oxford, 1982.
[14]	N. Bohr, “Philosophical Writings of Niels Bohr,” Ox Bow Press, Woodbridge, 1987.
[15]	P. A. M.Dirac, “The Principles of Quantum Mechanics,” 4th Edition, Clarendon Press, Oxford, 1958, p.255.
[16]	F. A. Jenkins and H. E. White, “Fundamentals of Optics,” 4th Edition, McGraw-Hill, Boston, 1957.
[17]	R. Feynmann, “Lectures on Physics Vol. 1,” Addison- Wesley, Boston, 1965.
[18]	B. I. Bleaney and B. Bleaney, “Electricity and magnetism,” Clarendon Press, Oxford, 1965.
[19]	MIT Electromagnetic Equation Propagation. http://ab-initio.mit.edu/~meep/meep.pdf
[20]	A. Einstein and H. Minkowski, “The Principle of Relativity,” The University of Calcutta, Kolkata, 1920.
[21]	J. H. Beaumont, A. J. Bourdillon and J. Bordas, “Optical Properties of PbI2 and PbF2,” Journal of Physics C, Vol. 10, No. 5, 1977, pp. 76l-771.
[22]	A. J. Bourdillon, “Optical Properties of Ionic Solids Using Synchrotron Radiation,” Oxford University Press, Oxford, 1976.
[23]	T. P. Cheng and L. F. Li, “Gauge Theory of Elementary Particle Physics,” Oxford University Press, Oxford, 2006.
[24]	M. Kaku, “Introduction to Superstrings and M-Theory; and Strings, Conformal Fields and M-Theory,” Springer, Berlin, 1999 and 2000.
[25]	I. S. Gradshteyn and I. M. Ryzhik, “Table of Integral, Se- ries, and Products,” Academic Press, Cambridge, 1980.
[26]	D. H. Tomboulian and P. L. Hartman, “Properties of Syn- chrotron Radiation,” Physical Review, Vol. 102, No. 6, 1956, pp. 1423-1447. doi:10.1103/PhysRev.102.1423
[27]	R. F. Egerton, “Electron Energy-Loss Spectroscopy in the Electron Microscope,” 3rd Edition, in Press.
[28]	P. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whelan, “Electron Microscopy of Thin Crystals,” Kreiger, Malabar, 1977.