On application of the Snyder theory to macroscopic objects
- PDF / 86,781 Bytes
- 2 Pages / 612 x 792 pts (letter) Page_size
- 60 Downloads / 236 Views
n Application of the Snyder Theory to Macroscopic Objects1 B. Kostenko Joint Institute for Nuclear Research, Dubna, 141980 Russia e-mail: [email protected] Abstract—Unitary representations of SO(4, 1) are used for construction of quantum gravitational wave functions of macroscopic bodies. Feasible experimental verification of the theory are pointed out. DOI: 10.1134/S1063779618010252
INTRODUCTION Recently a physical interpretation of Lie algebra of de Sitter group of quantum space-time is given for the Snyder theory [1], and, as a consequence, a possible manifestation of quantum gravitational effects in behavior of celestial bodies is discussed [2]. Here we discuss some mathematical questions omitted in the previous paper. First of all, is the Snyder theory applicable to macroscopic objects? In more technical terms: is really there a group contraction SO(4,1) → ISO(3,1) describing transition from de Sitter to Poincaré momentum space for objects with masses m @ mP ? It is also necessary to give a glance at contraction of unitary representations SO(4,1) → ISO(3,1) as far as namely them support the probability interpretation of measurable values corresponding to generators of de Sitter group.
take on form
2. TRANSFORMATION OF SNYDER MOMENTUM SPACE INTO POINCARÉ ONE Let us consider the four-dimensional stereographic projection [3] from the de Sitter hyper-sphere in the Snyder theory,
X λ → x λ = im p ∂ λ , ∂p and the de Sitter algebra turns into the Poincaré one for generators J μν and x λ . It is seen that transition from quantum to classical space has occurred here due to a large value of physical mass inserted in the momentum de Sitter space, as contrasted to common wisdom where macroscopic properties arise in the limit l P = 1 m p → 0 [5].
⎛ ⎛ p2 ⎞ ∂ ⎞ , J λ4 = i ⎜ hλλ 1 p λ p μ ∂ μ − mP ⎜1 + 2⎟ λ⎟ 2mP ∂p ⎝ ⎝ 4 mP ⎠ ∂ p ⎠ where hλμ = diag(1, − 1, − 1, − 1). To fulfil the de Sitter–Poincaré contraction we follow the general Wigner–Inönü procedure [4]: define a big parameter, R = p 2 4mP2 = m 2 4mP2 , and take the formal limit R → ∞. Let us introduce operator X λ = R −1J 4λ and rewrite the de Sitter algebra as follows i[J λμ, J ρσ ] = hλρ J μσ − hμρ J λσ + hμσ J λρ − hλσ J μρ, i[J λμ, X ν ] = hλν X μ − hμν X λ, i[ X μ, X ν ] = R −2 J μν. Now it is easy to check that in the limit R → ∞, when m @ mP and m @ pi2 2m , we have
(π 0 ) 2 − (π1) 2 − (π 2 )2 − (π3 )2 − (π 4 )2 = −mP2 , into a target pseudo-Euclidean momentum space:
π μ = Ω( p) p μ, μ = 0,1,2,3, π 4 = −mP Ω( p)(1 + p 2 4m 2p ), where Ω( p) =
⎛ ⎜1 − ⎝
p
2
4mP2
⎞ ⎟ ⎠
−1
3. ON UNITARY REPRESENTATIONS OF SO(4,1) It is known that eigenvalues of coordinate operator, xˆi = J 4i mP , in the Snyder theory are discrete, and time operator, tˆ = J 40 mP , has a continuous spectrum [5]. Using this fact, we can identify among the unitary representations of the de Sitter group those which may describe states with definite values of t and x (they are different as far as [ xˆi , tˆ] ≠ 0 ). One more reasonable assumption is the macroscopic particle has
, p = m . In
Data Loading...
