Double Special Relativity and Compton Effect

Double Special Relativity and Compton Effect

Corrado Massa

Via Fratelli Manfredi 55

42124 Reggio Emilia ( Italy )

ABSTRACT:

The Compton effect is studied in the framework of the generalized special relativity theory of MaguejoSmolin (the so called “double special relativity” with fundamental length L ).

The wavelength shift turns out to be identical with the usual Compton’s shift

( h / M c ) (1 – cos U) while the angular relation turns out to be L – dependent, and (at variance with the usual Compton effect) the scattering angles are quantized.

This work is dedicated to Meredith Kercher, murdered bya burglar in Perugia,Italy,1st November 2007.It is also dedicated to Amanda Knox and Raffaele Sollecito, innocent victims of the Italian judicial system. Please see , a site detailing the wrongful conviction of Amanda and Raffaele, and references therein.

1 . Dynamics of a particle in DSR.

The idea of a minimal length (quantum of space) rose in the 1930s to avoid divergences in quantum field theory and had a recent revival in connection with quantum gravity [ 1 ]. The idea

conflicts with the special relativity theory (shortly SR) because according to SR if the minimal length is L in an inertial frame S, the Lorentz contraction implies it is L’ = L / g ( g = the Lorentz factor) in another frame S’. However, L is expected to be a fundamental constant of nature related to the physical structure of space, and therefore (for the relativity principle) independent of the reference frame. To overcome such difficulty, Maguejio and Smolin suggest a dynamical solution: they modify the action of the Lorentz group so that the energy scale c h / L ( c = the velocity of light, h = the Planck constant) is left invariant [ 1 , 2, 3 ] . Their approach (often called Double Special Relativity, or Deformed Special Relativity, shortly DSR ) allows the complete equivalence of inertial frames, as demanded by the principle of relativity, but implies a departure from the usual special relativistic dynamics. The fundamental dynamic relation in DSR for a particle ( t = time coordinate, x, y, z = space coordinates) is

Pt 2 – P x2 – Py2 – Pz2 = M2 ( 1 – L Pt ) 2 ( 1 )

where Pt,Px, Py, Pz are the components of the energy – momentum 4 – vector, M is the rest (invariant) mass of the particle, and natural units ( c = h = 1 ) are employed. Eq ( 1 ) reduces to the usual special relativistic relation if L = 0.

It is easy to prove that Eq ( 1 ) implies the following transformation law for the energy E = Pt and the linear momentum of the particle ( we assume the usual standard configuration with V parallel to the x – axis, where V is the velocity of S relative to S’ )

E = b g ( E’ + V P’ ) ( 2 )

P = b g ( P’ + V E’ ) ( 3 )

b = [ 1 – L ( g – 1 ) E’ + L g V P’ ] – 1 ( 4 )

g = ( 1 – V 2 ) – 1 / 2 ( 5 )

e = M ( 1 + M L ) – 1 ( 6 )

where e = the rest energy of the particle, Py = P ’y , Pz = P’ z and P x = P for brevity.

If L = 0 we recover the usual SR transformation law and the usual mass – energy relation for a particle at rest, e = M (namely e = M c 2 if c is displayed).

If the particle is at rest in S’, then P’ = 0 , E’ = e , and the energy E and the linear momentum P as measured in the laboratory frame S are

E = b g e ( 7 )

P = b g e V ( 8 )

where V is the velocity of the particle with respect to S, and

b = [ 1 + L e ( g – 1 ) ] – 1 ( 9 )

Eq ( 5 ) can be written in the form

V = [ 1 – ( 1 / g )2 ] 1 / 2 ( 5 a )

Therefore eq ( 8 ) can be written as

P = b e ( g 2 – 1 ) 1 / 2 ( 10 )

The kinetic energy of the particle,

K = E – e , ( 11 )

for Eq ( 7 ) reads

K = e ( b g – 1 ) ( 12 )

2. Compton Effect in the DSR.

In SR the collision of two particles A and B is described by the law

A K + B K = A* K + B* K ( k = 0, 1, 2, 3 ) ( 13 )

where A k and B k are respectively the energy – momentum 4 – vector of A and B, and

unasterisked / asterisked variables refer to the 4 – moments before / after the collision.

In DSR the 2 – particle collision formula is more complex than Eq( 13 ) and reads [ 2 , 3 ]

( A ) A K + ( B ) B K = ( A* ) A* K + ( B* ) B* K ( 14 )

where for brevity ( A ) = 1 / ( 1 – L A t ) , ( A* ) = 1 / ( 1 – L A* t) and likewise for ( B ) and ( B* ).

The Compton effect involves the collision between a zero rest mass particle A (with frequency F, energy h F and linear momentum h F / c if c and h are displayed) and a particle B with rest mass M > 0.

From the equations of Section 1 we get ( with c = h = 1 )

A K = ( F , F , 0 , 0 ) ( 15 )

B K = ( e , 0 , 0 , 0 ) ( 16 )

A*K = ( F* , F* cos U , F* sin U , 0 ) ( 17 )

B* K = [ b g e , b e ( g 2 – 1 ) 1 / 2 cos W , e b ( g 2 – 1 ) 1 / 2 sin W , 0 ] ( 18 )

where U = the angle between the initial and final direction of A , W = the angle between the initial direction of A and the final direction of B (if I am not sufficiently clear, please see fig 3 , p. 25 of reference [ 4 ], where U and W are respectively named with the greek letter theta and theta primed respectively).

A little algebra gives

N – N* + M = M g ( 19 )

N – N* cos U = M ( g 2 – 1 )1 / 2 cos W ( 20 )

N* sin U = M ( g 2 – 1 ) 1 / 2 sin W ( 21 )

with N = F / ( 1 – L F ) , N* = F* / ( 1 – L F* ) .

Eq ( 19 ) gives

g = 1 + ( N / M ) – ( N* / M ) ( 22 )

which in turn gives

M2 ( g 2 – 1 ) = N 2 + N* 2 - 2 N N * + 2 M ( N – N* ) ( 23 )

which for eq ( 21 ) leads to

N 2 + N* 2 – 2 N N* + 2 M ( N – N* ) = N* 2 ( sin U / sin W ) 2 ( 24 )

Consider now Eq ( 20 ), it gives

sin 2 W = 1 – ( N – N* cos U )2 [ M2 ( g 2 – 1 )] – 1 ( 25 )

which for Eq( 24 ) gives

( N sin U )2 = M2 ( g 2 – 1 ) – ( N – N* cos U ) 2 ( 26 )

This and Eq ( 25 ) implies

F* [ 1 + ( F / M ) ( 1 – cos U ) ] = F ( 27 )

Display c and h, remember F = c / Z where Z = the wavelength of the zero rest mass particle A, and obtain

Z * – Z = ( h / M c ) ( 1 – cos U ) ( 28 )

Surprisingly, this is just the usual Compton formula for the wavelength shift.

Consider now Eq ( 20 ) and Eq ( 21 ); dividing Eq ( 20 ) by Eq ( 21 ) one finds ( with c = h = 1 )

( 1 – F L ) F* sin U = [ (1 – F* L) F – (1 – F L ) F* cos U ] tg W ( 29 )

If we substitute for F* the expression ( 27 ) we get

( 1 – F L ) cotg ( U / 2 ) = [ 1 – F L + ( F / m ) ] tg W ( 30 )

which reduces to the usual Compton angular relation if L = 0.

Come back now to Eq ( 28 ) and remember the constraint

Z * – Z = J L ( J = 0, 1 , 2 … ) ( 31 )

due to space quantization with L = minimal length = quantum of space.

From Eq ( 31) and Eq ( 28) we get

cos U = 1 – J L M ( = 1 – J L M c / h with displayed c and h ) ( 32 )

and therefore cotg ( U / 2 ) = [ ( 2 – J L M ) / J L M ] 1 / 2 which for Eq ( 30 ) implies

tg W = ( 1 – F L ) [ 1 – F L + ( F / M ) ] – 1 [ ( 2 /J L M ) – 1 ] 1 / 2 ( 33 )

or (with c and h displayed )

tg W = ( 1 – L / Z)( 1 – L / Z + h / L M c ) – 1 [ 2 h / J M c L – 1 ] 1 / 2 ( 33 a )

According to Eq (32 ) and Eq (33) the scattering angles U and W cannot have a continuous spectrum of values. This result is at variance with the usual Compton theory.

.

REFERENCES

[ 1 ] Sabine Hossenfelder, “ Minimal length scale scenarios for quantum gravity “ ,

Living Rev. Relativity , vol. 16, p. 2, year 2013. See also:

[ 2 ] Joao Maguejo Lee Smolin, “ Lorentz invariance with an invariant energy scale “

arXiv: hep – th / 0112090v2, 18 Dec 2001

[ 3 ] Joao Maguejo Lee Smolin, “ Generalized Lorentz invariance with an invariant energy scale “

arXiv: gr – qc / 0207085v1, 22 Jul 2002.

[ 4 ] Enrico Persico, Fundamentals of Quantum mechanics, Prentice – Hall, Inc. New York 1950.