# New fundamental Light Particle and breakdown of Stefan-Boltzmann's law.

1 Introduction

First, the quantization scheme for the local electromagnetic field in vacuum was presented by Planck in his black body radiation studies [1]. In this context, the classical Maxwell equations lead to appearance of the so-called ultraviolet catastrophe; to remove this problem, Planck proposed the model of the electromagnetic field as an ideal Bose gas of massless photons with spin one. However, Dirac [2] showed the Planck photon-gas could be obtained through a quantization scheme for the local electromagnetic field, presenting a theoretical description of the quantization of the local electromagnetic field in vacuum by use of a model Bose-gas of local plane electromagnetic waves propagating by speed c in vacuum.

In a different way, in regard to Plank and Dirac's models, we consider the structure of the electromagnetic field [3] as a non-ideal gas consisting of N neutral Light Bose Particles with spin 1 and finite mass m, confined in a box of volume V. The form of potential interaction between Light Particles is defined by introduction of the principle of wave-particle duality of de Broglie [4] and principle of gauge invariance. In this respect, a non-ideal Bose-gas consisting of Light Particles with spin 1 and non-zero rest mass is described by Planck's gas of massless photons together with a gas consisting of Light Particles in the condensate. In this context, we defined the Light Particle by the model of hard sphere particles [5]. Such definition of Light Particles leads to cutting off the spectrum of the electromagnetic wave by the boundary wave number [k.sub.0] = mc/[??] or boundary frequency [[omega].sub.[gamma]] = [10.sup.18] Hz of gamma radiation at the value of the rest mass of the Light Particle m = 1.8 x [10.sup.-4][m.sub.e]. On the other hand, the existence of the boundary wave number [k.sub.0] = mc/[??] for the electromagnetic field in vacuum is connected with the characteristic length of the interaction between two neighboring Light Bosons in the coordinate space with the minimal distance d = [1/[k.sub.0]] = [[??]/[m.sub.c]] = 2 x [10.sup.-9]m. This reasoning determines the density of Light Bosons N/V as [N/V] = [3/4[pi][d.sup.3]] = 0.3 x [10.sup.26][m.sup.-3].

It is well known that Stefan-Boltzmann's law [6] for thermal radiation, presented by Planck's formula [1], determines the average energy density U/V as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [??] is the Planck constant; [sigma] is the Stefan-Boltzmann

constant; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the average number of photons with the wave vector [??] at the temperature T:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Obviously, at T = 0, the average energy density vanishes in Eq.(1), i.e. U/V = 0, which follows from Stefan-Boltzmann's law.

However, as we show, the existence of the predicted Light Particles breaks Stefan-Boltzmann's law for black body radiation at low temperature.

2 Breakdown of Stefan-Boltzmann's law

Now, we consider the results of letter [3], where the average energy density of black radiation U/V is represented as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where m[c.sup.2][N.sub.0, T]/V is a new term, in regard to Plank's formula (1), which determines the energy density of Light Particles in the condensate; [N.sub.0, T]/V is the density of Light Particles in the condensate.

In this respect, the equation for the density of Light Particles in the condensate [N.sub.0, T]/V represents as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

with the real symmetrical function [L.sub.[??]] from the wave vector [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Our calculation shows that at absolute zero the value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and therefore the average energy density of black radiation U/V reduces to

[U/V] = [m[c.sup.2][N.sub.0, T] = 0/V] = [m[c.sup.2]N/V] - [[m.sup.4][c.sup.5]B(2, 3)/4[[pi].sup.2][h.sup.3]] [approximately equals] [m[c.sup.2]N/V], (6)

where B(2, 3) = [[integral].sup.1.sub.0]x[(1 - x).sup.2]dx = 0.1 is the beta function.

Thus, the average energy density of black radiation U/V is a constant at absolute zero. In fact, there is a breakdown of Stefan-Boltzmann's law for thermal radiation.

In conclusion, it should be also noted that Light Bosons in vacuum create photons, while Light Bosons in a homogeneous medium generate the so-called polaritons. This fact implies that photons and polaritons are quasiparticles, therefore, Bose-Einstein condensation of photons [7], polaritons [8] and exciton polaritons [9] has no physical sense.

Acknowledgements

This work is dedicated to the memory of the Great British Physicist Prof. Marshall Stoneham, F.R.S., (London Centre for Nanotechnology, and Department of Physics and Astronomy University College London, UK), who helped us with English. We are very grateful to him.

Submitted on January 6, 2011 / Accepted on January 25, 2011

References

[1.] Planck M. On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik, 1901, v. 4, 553-563.

[2.] Dirac P. A. M. The Principles of Quantum Mechanics. Clarendon Press, Oxford, 1958.

[3.] Minasyan V. N., Samoilov V. N. New resonance-polariton Bose-quasiparticles enhances optical transmission into nanoholes in metal films. Physics Letters A, 2011, v.375, 698-711.

[4.] de Broglie L. Researches on the quantum theory. Annalen der Physik, 1925, v. 3, 22-32.

[5.] Huang K. Statistical Mechanics. John Wiley, New York, 1963.

[6.] Stefan J. Uber die Beziehung zwischen der Warmestrahlung und der Temperatur. In: Sitzungsberichte der mathematisch naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften, Bd. 79 (Wien 1879), 391-428.

[7.] Klaers J., Schmitt J., Vewinger F., Weitz M. Bose-Einstein condensation of photons in an optical microcavity. Nature, 2010, v. 468, 545548.

[8.] Balili R., Hartwell V., Snoke D., Pfeiffer L., West K. Bose-Einstein condensation of microcavity polaritons in a trap. Science, 2007, v. 316, 1007-1010.

[9.] Kasprzak J. et al. Bose-Einstein condensation of exciton polaritons. Nature, 2006, v. 443, 409-414.

Vahan Minasyan and Valentin Samoilov

Scientific Center of Applied Research, JINR, Dubna, 141980, Russia

E-mails: mvahan@scar.jinr.ru; scar@off-serv.jinr.ru

First, the quantization scheme for the local electromagnetic field in vacuum was presented by Planck in his black body radiation studies [1]. In this context, the classical Maxwell equations lead to appearance of the so-called ultraviolet catastrophe; to remove this problem, Planck proposed the model of the electromagnetic field as an ideal Bose gas of massless photons with spin one. However, Dirac [2] showed the Planck photon-gas could be obtained through a quantization scheme for the local electromagnetic field, presenting a theoretical description of the quantization of the local electromagnetic field in vacuum by use of a model Bose-gas of local plane electromagnetic waves propagating by speed c in vacuum.

In a different way, in regard to Plank and Dirac's models, we consider the structure of the electromagnetic field [3] as a non-ideal gas consisting of N neutral Light Bose Particles with spin 1 and finite mass m, confined in a box of volume V. The form of potential interaction between Light Particles is defined by introduction of the principle of wave-particle duality of de Broglie [4] and principle of gauge invariance. In this respect, a non-ideal Bose-gas consisting of Light Particles with spin 1 and non-zero rest mass is described by Planck's gas of massless photons together with a gas consisting of Light Particles in the condensate. In this context, we defined the Light Particle by the model of hard sphere particles [5]. Such definition of Light Particles leads to cutting off the spectrum of the electromagnetic wave by the boundary wave number [k.sub.0] = mc/[??] or boundary frequency [[omega].sub.[gamma]] = [10.sup.18] Hz of gamma radiation at the value of the rest mass of the Light Particle m = 1.8 x [10.sup.-4][m.sub.e]. On the other hand, the existence of the boundary wave number [k.sub.0] = mc/[??] for the electromagnetic field in vacuum is connected with the characteristic length of the interaction between two neighboring Light Bosons in the coordinate space with the minimal distance d = [1/[k.sub.0]] = [[??]/[m.sub.c]] = 2 x [10.sup.-9]m. This reasoning determines the density of Light Bosons N/V as [N/V] = [3/4[pi][d.sup.3]] = 0.3 x [10.sup.26][m.sup.-3].

It is well known that Stefan-Boltzmann's law [6] for thermal radiation, presented by Planck's formula [1], determines the average energy density U/V as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [??] is the Planck constant; [sigma] is the Stefan-Boltzmann

constant; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the average number of photons with the wave vector [??] at the temperature T:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Obviously, at T = 0, the average energy density vanishes in Eq.(1), i.e. U/V = 0, which follows from Stefan-Boltzmann's law.

However, as we show, the existence of the predicted Light Particles breaks Stefan-Boltzmann's law for black body radiation at low temperature.

2 Breakdown of Stefan-Boltzmann's law

Now, we consider the results of letter [3], where the average energy density of black radiation U/V is represented as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where m[c.sup.2][N.sub.0, T]/V is a new term, in regard to Plank's formula (1), which determines the energy density of Light Particles in the condensate; [N.sub.0, T]/V is the density of Light Particles in the condensate.

In this respect, the equation for the density of Light Particles in the condensate [N.sub.0, T]/V represents as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

with the real symmetrical function [L.sub.[??]] from the wave vector [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Our calculation shows that at absolute zero the value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and therefore the average energy density of black radiation U/V reduces to

[U/V] = [m[c.sup.2][N.sub.0, T] = 0/V] = [m[c.sup.2]N/V] - [[m.sup.4][c.sup.5]B(2, 3)/4[[pi].sup.2][h.sup.3]] [approximately equals] [m[c.sup.2]N/V], (6)

where B(2, 3) = [[integral].sup.1.sub.0]x[(1 - x).sup.2]dx = 0.1 is the beta function.

Thus, the average energy density of black radiation U/V is a constant at absolute zero. In fact, there is a breakdown of Stefan-Boltzmann's law for thermal radiation.

In conclusion, it should be also noted that Light Bosons in vacuum create photons, while Light Bosons in a homogeneous medium generate the so-called polaritons. This fact implies that photons and polaritons are quasiparticles, therefore, Bose-Einstein condensation of photons [7], polaritons [8] and exciton polaritons [9] has no physical sense.

Acknowledgements

This work is dedicated to the memory of the Great British Physicist Prof. Marshall Stoneham, F.R.S., (London Centre for Nanotechnology, and Department of Physics and Astronomy University College London, UK), who helped us with English. We are very grateful to him.

Submitted on January 6, 2011 / Accepted on January 25, 2011

References

[1.] Planck M. On the Law of Distribution of Energy in the Normal Spectrum. Annalen der Physik, 1901, v. 4, 553-563.

[2.] Dirac P. A. M. The Principles of Quantum Mechanics. Clarendon Press, Oxford, 1958.

[3.] Minasyan V. N., Samoilov V. N. New resonance-polariton Bose-quasiparticles enhances optical transmission into nanoholes in metal films. Physics Letters A, 2011, v.375, 698-711.

[4.] de Broglie L. Researches on the quantum theory. Annalen der Physik, 1925, v. 3, 22-32.

[5.] Huang K. Statistical Mechanics. John Wiley, New York, 1963.

[6.] Stefan J. Uber die Beziehung zwischen der Warmestrahlung und der Temperatur. In: Sitzungsberichte der mathematisch naturwissenschaftlichen Classe der kaiserlichen Akademie der Wissenschaften, Bd. 79 (Wien 1879), 391-428.

[7.] Klaers J., Schmitt J., Vewinger F., Weitz M. Bose-Einstein condensation of photons in an optical microcavity. Nature, 2010, v. 468, 545548.

[8.] Balili R., Hartwell V., Snoke D., Pfeiffer L., West K. Bose-Einstein condensation of microcavity polaritons in a trap. Science, 2007, v. 316, 1007-1010.

[9.] Kasprzak J. et al. Bose-Einstein condensation of exciton polaritons. Nature, 2006, v. 443, 409-414.

Vahan Minasyan and Valentin Samoilov

Scientific Center of Applied Research, JINR, Dubna, 141980, Russia

E-mails: mvahan@scar.jinr.ru; scar@off-serv.jinr.ru

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Author: | Minasyan, Vahan; Samoilov, Valentin |
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Publication: | Progress in Physics |

Date: | Apr 1, 2011 |

Words: | 1038 |

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