Are relations P=hk and E=mc2 simultaneously compatible

with principle of wave-particle duality?

V.G. Veselago (v.veselago@relcom.ru)

A.M. Prokhorov General Physics Institute, 119991, Moscow, Russia

Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Russia

With the advent in scientific language the terms «negative refraction coefficient» or «negative refraction» as 

well as related terms, researchers put some questions, which generally speaking have a much broader 

significance than the above words. The notion of «negative refraction coefficient» was shown for the first 

time probably in our early work1

. However, we have shown2 that a simple substitution of negative values n 

in some formula of electrodynamics and optics often leads to grave mistakes. Moreover, if the phase and 

group velocity are antiparallel, you must be very careful with some other formulas, which are not directly 

included value of n. This applies, for example, to the very well known formula P=hk, linking the value of the 

photon linear momentum with its wave vector. It is obvious that in the case of negative wave vector k, this 

formula gives a negative value for the photon linear momentum, and thus in case of the light absorption light 

pressure should be replaced by light attraction. This is all the more imperative that, strangely enough, the 

value of the photon linear momentum, even for the usual matter with positive n case is a subject of debate

for over 100 years3

. This discussion is based on the existence of two completely different energy-momentum 

tensors for electromagnetic fields, namely tensors in Minkowski form and in the form of Abraham. They 

give different expression for the forces in the transparent body by passing it through the electromagnetic 

radiation. Interesting to note that the tensor of Abraham in fact is not a tensor, because it is not a relativistic 

invariant. As to the Minkowski tensor, its use leads to some unusual expression M=E / vgrvph for the 

masses, transferred by the radiation from the emitter to the receiver (values vph and vgr are phase and group 

velocities accordingly). From this expression becomes clear, that in the case of negative refractive index the 

mass is transferred not from the radiator to the receiver, but in opposite direction from the receiver to the 

radiator The above considerations indicate that relations P=hk and E=mc

2

are not simultaneously 

compatible with wave-particle duality principle4

.

This work was supported by Russian Foundation for Basic Research, grants NoNo 06-02-16830-а, 07-02-00233-а, 09-02-01186-a and 09-02-01519-a

References 1 V.G.Veselago Electrodynamics of substancies with simultaneosly negative values of electric and magnetic permeabilities. 

Sov.Phys.Usp., 10, pp.509-514, (1968) 2 V.Veselago, L.Braginsky, V.Shklover, C.Hafner. Negative refractive index materials. Journal of Computation and 

Theoretical Nanoscience, 3, pp.1-30, (2006) 3 Robert N. Pfeifer, Timo A. Nieminen, Norman R. Heckenberg, and Halina Rubinsztein-Dunlop. Momentum of an

electromagnetic wave in dielectric media. Rev.Mod.Phys. 79, 1197-1216 (2007) 4 V G Veselago.Energy, linear momentum, and mass transfer by an electromagnetic wave in a negative-refraction medium. 

Physics Uspekhi 52, 649 – 654, (2009)