Approaches to Quantum Gravity

522 F. Girelliwhere λ i are Lagrange multipliers implementing the two first class constraints, Mis a constant that will specify the mass m. In order to recover a 4d particle, we needto introduce a gauge fixing C that will allow us to reduce the ten dimensional phasespace ( y A , P A)to an eight dimensional one(x μ , P μ)together with the constraintH = P A P A + M P 2 [11]. The symplectic form on the eight dimensional phase spaceis not arbitrary anymore but given by the the Dirac bracket1{φ,ψ} D ={φ,ψ}−{φ,C}{H, C} {H,ψ}+{φ, H} 1{H, C} {C,ψ},where ψ, φ, are functions on phase space and H is the constraint, such that{C, H} ̸= 0. The reduced phase space coordinates are determined such that theycommute with both the constraint and the gauge fixing. Note also that P can beinterpreted as a coordinates system on the de Sitter space defined by the 5d massshell condition dS ∼{P A , P A P A =−M P 2 }.As a first example, we can introduce the gauge fixing C = y A π A − T .Itisthen easy to see that a choice of physical coordinates is just given by the SnydercoordinatesP μ ≡ M PP μP 4, x μ = M PJ μ4 ≡ M P(y μ P 4 − y 4 P μ ). (26.8)The symplectic form is the Snyder symplectic form{x μ , x ν }=( M P) 2J μν , {x μ , P ν }=(η μν − P μP νM P2). (26.9)The physical mass m can be determined solely from the constants in the actionnamely, M, M P , which both have dimension mass.P 2 = m 2 2= M M2 2− M PP .M 2The rest mass m is bounded by M P sinceweneedM 2 − MP 2 ≥ 0.A different gauge fixing C = y 0−y 4P 0 −P 4− T , provides the bicrossproduct basis [9].The physical phase space variables areP 0 ≡ M P ln P 4 − P 0M P, P i ≡ M PP iP 0 − P 4,x 0 ≡ M PJ 40 , x i ≡ M P(J i0 − J i4 ), (26.10)on the domain P 4 − P 0 > 0. These variables encode the so called κ-Minkowskisymplectic structure on the 4d phase space,{x 0 , P 0 }=1, {x i , P j }=−δ ij , (26.11){x 0 , x i }= + 1 M Px i , {x 0 , P i }=− 1 M PP i , (26.12)