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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) 103–108 105 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) Electronic Journal of Theoretical Physics 10, No. 29 (2013) 103–108 106 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