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On gravitational fluctuations of quantum vacuum interacting with photons
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 51–52 www.elsevier.de/ijleo
On gravitational fluctuations of quantum vacuum interacting with photons M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/Julio Palacios 11, 28029-Madrid, Spain Received 17 April 2006; accepted 18 June 2006
Abstract An upper bound for the photon mass is calculated by regarding photons under the action of a gravitational field within the framework of fluctuations in the quantum vacuum. Furthermore, some considerations upon zero-point energy are made in order to estimate interaction range in terms of distance. r 2006 Elsevier GmbH. All rights reserved. Keywords: Photon mass; Quantum gravitational fluctuations; Zero-point energy
The fact that the total strength of the interaction of a photon with the quantum vacuum is proportional to the value of the Planck constant [1] has a great relevance in order to estimate photon rest-mass which, in the following, will be also denominated briefly photon mass. An upper bound for this mass has been determined by means of a zero-point energy approach [1]. In the present note, we shall find the supremum of the photon mass as a function of wavelength by assuming a single photon under the action of the gravitational field created by the Sun [2]. Firstly, we state that the rest-mass energy of a photon under the above gravitational field is majorized by the potential energy of the photon due to this field; then, we have c2 m0 ðlÞ
GMmðlÞ r
(1)
(for the meaning of the quantities involved in formula (1), we refer to [2]). Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.06.009
Since relativistic mass mðlÞ is given by mðlÞ ¼ h=ðclÞ , inequality (1) becomes m0 ðlÞp
GMh , c3 lr
(2)
so that, by inspecting expression (2), it is clear that there exists a linear relationship between the Planck constant and the driving energy of the involved quantum fluctuations [1]. In addition, formula (2) was obtained in Ref. [2] by considerations of energy conservation. Now we calculate the supremum of the function m0 ðlÞ on the left-hand side of formula (2) by minimizing the right-hand side of the formula in question (fixing distance). To get this end, by assuming more realistic data than in Ref. [1], we take lmax 104 m so that, from relation (2) it follows that sup m0 ðlÞ 5:2 1049 g which is in good agreement with one of the two values measured in [3] namely 3:5 1049 g approximately. Next, we will state the following relationship where the left-hand side represents the ground-state energy of a one-dimensional quantum harmonic oscillator [1]: hc GMh . 2l clr
(3)
ARTICLE IN PRESS 52
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 119 (2008) 51–52
Notice that formula (3) constitutes a zero-point energy approach. From this formula, distance is found to be of the order of magnitude of twice the product of the universal gravitation constant by the Sun mass divided by the squared speed of light in vacuum. In addition, by combining formulas (2) and (3), it follows that photon mass is majorized approximately by one-half the relativistic mass; this fact agrees with relation (2) of Ref. [1], as expected. In summary, we have performed a brief study upon the theoretical determination of the photon mass as a function of wavelength, giving new ideas with respect to previous work [1,2]. The concept of quantum fluctuation has been used and our calculations have given rise to formula (2) of Ref. [1], which can be deduced by establishing that the rest-mass energy of a photon is majorized by zero-point energy. Our result concerning
the supremum of photon mass is in excellent agreement with experimental data [3] improving substantially previous work [1,2]; at this point, let us note the relevance of the mass in question as a function of wavelength.
References [1] M.A. Grado-Caffaro, M. Grado-Caffaro, A zero-point energy approach for estimating the photon mass, Optik 117 (2006) 93–94. [2] M.A. Grado-Caffaro, M. Grado-Caffaro, Estimation of the photon rest-mass from gravitational interaction, Optik 116 (2005) 237–238. [3] R. Lakes, Experimental limits on the photon mass and cosmic magnetic vector potential, Phys. Rev. Lett. 80 (1998) 1826–1829.
Optik
Optics
Optik 119 (2008) 51–52 www.elsevier.de/ijleo
On gravitational fluctuations of quantum vacuum interacting with photons M.A. Grado-Caffaro, M. Grado-Caffaro SAPIENZA S.L. (Scientific Consultants), C/Julio Palacios 11, 28029-Madrid, Spain Received 17 April 2006; accepted 18 June 2006
Abstract An upper bound for the photon mass is calculated by regarding photons under the action of a gravitational field within the framework of fluctuations in the quantum vacuum. Furthermore, some considerations upon zero-point energy are made in order to estimate interaction range in terms of distance. r 2006 Elsevier GmbH. All rights reserved. Keywords: Photon mass; Quantum gravitational fluctuations; Zero-point energy
The fact that the total strength of the interaction of a photon with the quantum vacuum is proportional to the value of the Planck constant [1] has a great relevance in order to estimate photon rest-mass which, in the following, will be also denominated briefly photon mass. An upper bound for this mass has been determined by means of a zero-point energy approach [1]. In the present note, we shall find the supremum of the photon mass as a function of wavelength by assuming a single photon under the action of the gravitational field created by the Sun [2]. Firstly, we state that the rest-mass energy of a photon under the above gravitational field is majorized by the potential energy of the photon due to this field; then, we have c2 m0 ðlÞ
GMmðlÞ r
(1)
(for the meaning of the quantities involved in formula (1), we refer to [2]). Corresponding author.
E-mail address: [email protected] (M.A. Grado-Caffaro). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.06.009
Since relativistic mass mðlÞ is given by mðlÞ ¼ h=ðclÞ , inequality (1) becomes m0 ðlÞp
GMh , c3 lr
(2)
so that, by inspecting expression (2), it is clear that there exists a linear relationship between the Planck constant and the driving energy of the involved quantum fluctuations [1]. In addition, formula (2) was obtained in Ref. [2] by considerations of energy conservation. Now we calculate the supremum of the function m0 ðlÞ on the left-hand side of formula (2) by minimizing the right-hand side of the formula in question (fixing distance). To get this end, by assuming more realistic data than in Ref. [1], we take lmax 104 m so that, from relation (2) it follows that sup m0 ðlÞ 5:2 1049 g which is in good agreement with one of the two values measured in [3] namely 3:5 1049 g approximately. Next, we will state the following relationship where the left-hand side represents the ground-state energy of a one-dimensional quantum harmonic oscillator [1]: hc GMh . 2l clr
(3)
ARTICLE IN PRESS 52
M.A. Grado-Caffaro, M. Grado-Caffaro / Optik 119 (2008) 51–52
Notice that formula (3) constitutes a zero-point energy approach. From this formula, distance is found to be of the order of magnitude of twice the product of the universal gravitation constant by the Sun mass divided by the squared speed of light in vacuum. In addition, by combining formulas (2) and (3), it follows that photon mass is majorized approximately by one-half the relativistic mass; this fact agrees with relation (2) of Ref. [1], as expected. In summary, we have performed a brief study upon the theoretical determination of the photon mass as a function of wavelength, giving new ideas with respect to previous work [1,2]. The concept of quantum fluctuation has been used and our calculations have given rise to formula (2) of Ref. [1], which can be deduced by establishing that the rest-mass energy of a photon is majorized by zero-point energy. Our result concerning
the supremum of photon mass is in excellent agreement with experimental data [3] improving substantially previous work [1,2]; at this point, let us note the relevance of the mass in question as a function of wavelength.
References [1] M.A. Grado-Caffaro, M. Grado-Caffaro, A zero-point energy approach for estimating the photon mass, Optik 117 (2006) 93–94. [2] M.A. Grado-Caffaro, M. Grado-Caffaro, Estimation of the photon rest-mass from gravitational interaction, Optik 116 (2005) 237–238. [3] R. Lakes, Experimental limits on the photon mass and cosmic magnetic vector potential, Phys. Rev. Lett. 80 (1998) 1826–1829.