Optik 121 (2010) 2094–2095
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Photon velocity from the Klein–Gordon equation M.A. Grado-Caffaro , M. Grado-Caffaro C/ Julio Palacios 11, 9-B, 28029-Madrid, Spain
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Article history: Received 25 February 2009 Accepted 28 June 2009
An approximate mathematical relationship for the velocity of a photon as a non-zero rest-mass quantum particle is derived from the ﬁeld-free Klein–Gordon equation in the framework of the de Broglie–Bohm theory of quantum mechanics. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Photon velocity Photon mass Klein–Gordon equation De Broglie–Bohm theory
Problems relative to the mass of some quantum particles as, for example, the neutrino and the photon are certainly relevant since, for instance, it has been assumed that the neutrino rest-mass is zero (after the Standard Model) and now we know that this assumption is not true. Also, the rest-mass of the photon has been assumed to be zero, which constitutes an oversimpliﬁcation because the mass in question has been found to be (although very small) non-zero [1–4]. Certainly, issues related to the photon mass are very relevant from both the standpoints of particle physics and optics. The fact that the photon rest-mass is non-zero has signiﬁcant implications on high-energy physics. In particular, the above fact gives rise to that the speed of light depends on wavelength and to the existence of longitudinal light-waves . By the way, speed of light (photon velocity) will be examined in the following from the Klein–Gordon equation within the framework of the de Broglie–Bohm theory of quantum mechanics. We start from the ﬁeld-free Klein–Gordon equation for a single photon. This equation reads:
1 @2 C c2 @t 2
r ; tÞ is the associated wavefunction, m0 is the photon where Cð~ mass (which depends upon wavelength [2–4]), c is the speed of light in vacuum, and ‘ is the reduced Planck constant. Eq. (1) can be rewritten as follows:
‘ 2 r2 C ‘ 2 @2 C ¼ m0 c2 m0 C m0 c2 C @t 2
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(M.A. Grado-Caffaro). URL: http://www.sapienzastudies.com (M.A. Grado-Caffaro). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.06.009
Looking at Eq. (2), note that the right-hand side is the restmass energy of the photon and that the left-hand side can be viewed as the difference of two terms so that the ﬁrst term is the total energy while the second one represents the kinetic energy. In addition, notice that the above difference is sufﬁciently small. Now we wish to obtain an expression for the photon velocity as a non-zero rest-mass mass particle within the context of the de Broglie–Bohm theory of quantum mechanics [5–7]. To get this end, we may consider, for instance, the formula which gives the total energy namely: m0 c2 E ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 vc2
Equating relation (3) with the ﬁrst term of the left-hand side of Eq. (2), it follows: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u u m4 c4 C2 ð4Þ v ¼ ct1 4 0 2 2 ‘ ðr CÞ The subradical quantity in formula (4) can be approximated as follows: m4 c4 C2 m2 c 2 C m2 c2 C m2 c2 C 1 20 2 1 20 2 ð5Þ 1 4 0 2 2 ¼ 1 þ 20 2 ‘ ðr CÞ ‘ r C ‘ r C ‘ r C By inserting the approximation (5) into relation (4), one has: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m2 c2 C ð6Þ v c 1 20 2
‘ r C
In summary, we have found an expression (Eq. (4)) for the photon velocity as a function of the associated wavefunction over the space-time and its laplacian operator relative to the three spatial coordinates. Moreover, by means of a suitable
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 121 (2010) 2094–2095
approximation (relationship (5)) based upon the fact that the photon rest-mass is small enough (but non-zero), we have arrived at formula (6), which constitutes a useful representation of the photon velocity as a quantity which is sufﬁciently close to c. References  R. Lakes, Experimental limits on the photon mass and cosmic magnetic vector potential, Phys. Rev. Lett. 80 (1998) 1826–1829.
 M.A. Grado-Caffaro, M. Grado-Caffaro, Theoretical determination of the photon rest-mass, Optik 114 (2003) 142–143.  M.A. Grado-Caffaro, M. Grado-Caffaro, A qualitative estimation of the photon rest-mass and related topics, Optik 114 (2003) 239–240.  M.A. Grado-Caffaro, M. Grado-Caffaro, A zero-point energy approach for estimating the photon mass, Optik 117 (2006) 93–94.  D.Z. Albert, Quantum mechanics and experience, Harvard University Press, Cambridge, Mass., 1992.  P.R. Holland, The quantum theory of motion, Cambridge University Press, Cambridge, 1993.  A. Valentini, Pilot-wave theory: an alternative approach to modern physics, Cambridge University Press, Cambridge, 2001.