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EJTP 10, No. 29 (2013) 103–108
Electronic Journal of Theoretical Physics
Reciprocal Symmetric Kinematics and
Correspondence between Special Relativity and
Quantum Mechanics
Mushfiq Ahmad∗
Department of Physics, Rajshahi University, Rajshahi-6205, Bangladesh
Received 30 January 2013, Accepted 10 May 2013, Published 12 July 2013
Abstract:
We have defined reciprocal symmetry and observed its presence in Special
Relativity and quantum mechanics. Therefore, reciprocal symmetry relates Special Relativity
to quantum mechanics. Reciprocal symmetry implies that there is a boundary value velocity c,
which is the limiting value and which is also constant under addition. We have, then, developed
reciprocal symmetric law of addition of velocities. In case of collinear velocities it agrees with
Lorentz-Einstein law of addition. General reciprocal symmetric transformation is different
from General Lorentz transformation. Reciprocal symmetric transformation is associative and
complex. A comparison between RST and LT is given.
c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Reciprocal Symmetry; Reciprocal Symmetric Transformation; Lorentz
Transformation; Associativity
PACS (2010): 03.30.+p; 03.65.-w
1.
Introduction
Planck and Einstein: In 1900 Planck’s hypothesis set a lower limit to energy quantum.
In 1905 Einstein suggested velocity has an upper limit. If a quantity has an upper limit,
its reciprocal has a lower limit. This suggests a reciprocal relation between Planck’s
hypothesis and Einstein’s postulate.
Slowness: Corresponding to every velocity v = x/t, we can define slowness v =1/v =
t/x. Physics is independent of the way we describe it. The principle of objectivity demands that a description in terms of velocity should be as valid as the description in
terms of slowness. We intend to study how this velocity-slowness symmetry relates Special Relativity to quantum mechanics.
∗
Email:mushfi[email protected]
Electronic Journal of Theoretical Physics 10, No. 29 (2013) 103–108
104
Reciprocal Number: Every number (except 0) has a reciprocal. If we can establish isomorphism between numbers and their reciprocals, the description of an event in
terms of reciprocals of numbers should be as valid as a description in terms of numbers
themselves.
2.
Reciprocity between Relativity and Quantum Mechanics
From the velocity v we form the dimensionless quantity v/c where c is an arbitrarily chosen
(scale) velocity. We define the reciprocal velocity, u∗ , and slowness u corresponding to
velocity u by
c
u∗
u
= =
(1)
c
c
u
2.1 Reciprocity between Relativity and De Broglie Relation
Consider Einstein’s relation
mc2 = E = hν
(2)
And the wave length for a particle with velocity u is given by de Broglie relation
λ=
h
m.u
(3)
Combining (2) and (3), we get the velocity, u∗ for de Broglie wave
u∗ = λν =
hν
c2
=
m.u
u
(4)
2.2 Reciprocity between Einstein’s Postulate and Planck’s Hypothesis
Let Eu be 2 times the kinetic energy of a particle with velocity u
Eu = mu2
(5)
|u| ≤ c
(6)
According to Einstein’s postulate
Therefore, using (5) and (6) and replacing u by c
Eu ≤ Ec
Corresponding to (2) we define reciprocal energy
2 2
2
(Ec )2
(m.c2 )
c
∗
=
=m
= m. (u∗ )2 = Eu∗
Eu =
2
Eu
m.u
u
(7)
(8)
Therefore, corresponding to (7) we have
Eu ∗ ≥ Ec
(9)
Electronic Journal of Theoretical Physics 10, No. 29 (2013) 103–108
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We relate
Ec = hν
(10)
Where m will determine ν. By (9) Ec sets the lower limit of Eu ∗ just as by (7) it sets the
upper limit ofEu . Therefore, we may state Einstein’s postulate (and Planck’s hypothesis)
in two equivalent ways as below:
Einstein’s picture: When a body gains (or loses) energyEu the gain (or loss) cannot be
greater thanEc .
Planck’s picture: When a body gains (or loses) energy Eu ∗ the gain (or loss) cannot be
less thanEc .
In terms of speed the above statement will be:
Einstein’s picture: When a body gains (or loses) speed u the gain (or loss) of speed
cannot be greater than c.
Planck’s picture: When a body gains (or loses) reciprocal speed u∗ the gain (or loss) of
reciprocal speed cannot be less than c.
2.3 Discreteness
Consider two reciprocal energy levels Eu ∗ and Ev ∗. Let the difference between the reciprocal energy levels be Eu−v ∗. Since Eu−v ∗ is a reciprocal energy (group property), it
fulfills condition (9)
(11)
Eu−v ∗ ≥ Ec
The above relation shows that the difference between any two reciprocal energy levels
cannot be less than Ec which is a non-zero quantity. Therefore, reciprocal energy is
discrete. Similarly, let the difference between two reciprocal speed levels u∗ and v ∗ be
du−v ∗. By the requirement of group property du−v ∗ is also a reciprocal speed. Therefore,
by (1)
(12)
|du−v ∗ | ≥ c
Therefore, the minimum difference between two reciprocal speed levels is a non-zero
reciprocal speed. Therefore, reciprocal speeds are discrete quantities.
3.
Reciprocal Symmetry
To reflect the symmetry mentioned in sections 1 and 2, we postulate that the law of
addition (composition) of velocities ⊕ should fulfill the symmetry condition
Postulate:
u ⊕ v = (u∗ ) ⊕ (v ∗ )
(13)
(13) implies[1]
u ⊕ (v ∗ ) = (u∗ ) ⊕ v = (u ⊕ v) ∗ =
c2
u⊕v
(14)
(14) may also be written as
{u ⊕ (v ∗ )} . (u ⊕ v) = c2
(15)
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If v = c (1) shows v ∗ = c and (15) gives the invariance[2] under addition
c ⊕ v = ±c
(16)
(16) should be valid for al values of v including 0. Therefore,
c⊕v =c
(17)
4.
Motion in One Dimension
Two definitions of ⊕(⊕ and ⊕∗) fulfill requirement (13)
u+v
1 + u.v/c2
(17)
uv + c2
u+v
(18)
|u ⊕ v| ≤ c if |u| ≤ c and |v| ≤ c
(19)
|u∗ ⊕ v| ≥ c if |u| ≥ c and |v| ≥ c
(20)
u ⊕ v = (u∗ ) ⊕ (v ∗ ) =
And
u ⊕ ∗v = (u∗ ) ⊕ ∗ (v ∗ ) =
We choose ⊕ and ⊕∗ such that
And
(17) is Lorentz-Einstein law of addition of velocities and (20) gives the law of addition
of de Broglie velocities. Both of them fulfill requirement (15)
5.
General Lorentz Transformation and Reciprocity
The general Lorentz transformation is given by
w = u ⊕E (−v) =
Where
u−v
1 γu u × (u × v)
−
1 − u · v/c2 c2 γu + 1 1 − u · v/c2
γu = 1/ 1 − (u/c)2
(21)
(22)
Its properties under reciprocal inversion corresponding to (15) are
{u ⊕E v∗ } · {u ⊕E v} = c2 for v∗ · v = c2
(23)
{u∗ ⊕E v} · {u ⊕E v} = c2 for u∗ · u = c2
(24)
But
(21) does not have a symmetry property comparable to (15). (22) and (24) show
that Lorentz transformation is partially reciprocal symmetric, but it reveals a pathology
that Lorentz transformation does not treat the two velocities equally. This is against the
principle of relativity.
Electronic Journal of Theoretical Physics 10, No. 29 (2013) 103–108
6.
107
Reciprocal Symmetric Transformation
We want to replace u ⊕E v of (25) by[3] u ⊕RS v (E is replaced by RS ), which should
have property (15). Reciprocal symmetry will ensure2 (16).
wRS = u ⊕RS (−v) =
u − v − iu × v/c
1 − u · v/c2
(25)
We shall call u∗ a reciprocal of u if
u · u∗ = c2 and (u∗ )∗ = u
For arbitrary G
u∗ =
G + iG × u/c
and (u∗ )∗ = u
2
u · G/c
(26)
(27)
And corresponding to (15) we have the symmetry relation
wRS = u∗ ⊕RS (−v∗ ) = u ⊕RS (−v)
(28)
The reciprocal sum corresponding to (20) and for arbitrary G will be
G.c2 + (u · v) G − G × (u × v) + i.c.G × (u + v)
G · (u + v) + i.G · (u × v)/c
(29)
c2 + u · v
v=
{u + v + iu × v/c}
u2 + v2 + 2.u · v − (u × v/c)2
(30)
wRS = u∗ ⊕RS v =
Or, if G = u ⊕RS v
∗
wRS = u ⊕RS
Conclusion
We have found Einstein’s law of addition of velocities (17) from symmetry requirement
(13), (without invoking Einstein’s postulate). Reciprocal symmetry also relates Special
Relativity to quantum mechanics.
References
[1] M. Ahmad And M.O.G. Talukder. Phys. Essays 24, 4 (2011), p. 593. DOI:
10.4006/1.3658742
[2] M. Ahmad. J. Sci. Res. 1(32). 270-274 (2009). DOI. 10.3329.jsr.v1i2.1875
[3] M. Ahmad and M. S. Alam. Phys. Essays, 22, 2 (2009) 164-167. DOI:
10.4006/1.3114542