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Cosmology with the Compact Phase Space of Matter Homogeneous cosmology Perturbative inhomogeneities Cosmology with the compact phase space of matter Tomasz Trzesniewski´ ∗ Institute for Theoretical Physics, Wrocław University, Poland / Institute of Physics, Jagiellonian University, Poland September 26, 2017 ∗J. Mielczarek & T.T., Phys. Rev. D 96, 043522 (2017) T. Trzesniewski´ Compact phase space and cosmology Homogeneous cosmology Perturbative inhomogeneities Outline: 1 Homogeneous cosmological model Spherical phase space for the field Classical dynamics of the model 2 Perturbative quantum inhomogeneities Quantization of the linearized model First order corrections to the standard case T. Trzesniewski´ Compact phase space and cosmology 1 / 14 Homogeneous cosmology Perturbative inhomogeneities Outline: 1 Homogeneous cosmological model Spherical phase space for the field Classical dynamics of the model 2 Perturbative quantum inhomogeneities Quantization of the linearized model First order corrections to the standard case T. Trzesniewski´ Compact phase space and cosmology 1 / 14 Homogeneous cosmology Perturbative inhomogeneities Context and the existing work Momentum spaces, or phase spaces, with nontrivial geometry that appear in quantum gravity Born reciprocity and three-dimensional gravity Quantum gravity phenomenology, relative locality framework Group field theory, loop quantum cosmology etc. Target manifolds for field values of non-linear sigma models and the Tseytlin string action The principle of finiteness of physical quantities, at the base of the Born-Infeld theory Potential connections between quantum gravity, cosmology and condensed matter physics Nonlinear Field Space Theory: J. Mielczarek and T.T., Phys. Lett. B 759, 424 (2016) J. Mielczarek, Universe 3, 29 (2017) T.T., Acta Phys. Pol. B Proc. Suppl. 10, 329 (2017) J. Bilski, S. Brahma, A. Marcianò and J. Mielczarek, arXiv:1708.03207 [hep-th] T. Trzesniewski´ Compact phase space and cosmology 2 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Phase space variables Phase space Γ = R2 is formed by values of a scalar field ' and its conjugate momentum π' at every point of space Σ. We assume that Γ is actually a sphere, parametrized in terms of usual angles φ and θ or the spin-like vector S = (Sx ; Sy ; Sz ), so that π' ' Sx := S sin θ cos φ = S cos cos ; (1) R2 R1 π' ' Sy := S sin θ sin φ = S cos sin ; (2) R2 R1 π' Sz := S cos θ = S sin ; (3) R2 where R1, R2 are certain dimensionful constants and '=R1 2 [−π; π), π π π'=R2 2 [− 2 ; 2 ]. For a field defined on the Minkowski background we have the limiting condition R1R2 = S. T. Trzesniewski´ Compact phase space and cosmology 3 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Phase space algebra In the Minkowski case the symplectic form is given by the area 2-form π' !M = S sin θ dφ ^ dθ = cos dπ' ^ d'; (4) R2M R satisfying S2 !M = 4πS. For the FRW background we introduce the 3 gravitational field variable q ≡ V0a (here a is the scale factor and V0 a fiducial spatial volume) and its conjugate momentum p. Then we assume that the generalized symplectic total form is π' ! := dp ^ dq + cos dπ' ^ d': (5) R2(q) However, for ! to be a closed form we need R2(q) = R2. The corre- sponding Poisson bracket has the form @· @· @· @· 1 @· @· @· @· {·; ·} = − + π' − : (6) @q @p @p @q cos @' @π' @π' @' R2 T. Trzesniewski´ Compact phase space and cosmology 4 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Hamiltonian from the Heisenberg model The Hamiltonian of the continuous XXZ Heisenberg model coupled to a magnetic field B (for convenience, B := (Bx ; 0; 0)) has the form Z 3 ~ 2 2 2 HXXZ = − d x J (rSx ) + (rSy ) + ∆(rSz ) +µ ~ B · S ; (7) where J~, µ~ are coupling constants and ∆ the anisotropy parameter. The homogeneous field corresponds to the term / B, which we adapt to the FRW background multiplying the measure d 3x by Na3, obtaining HSmatmat = −Nq µ~Bx Sx (8) µ~Bx S 2 µ~Bx S 2 4−n n = Nq −µ~Bx S + 2 π' + 2 ' + O(' π') : 2R2 2R1 The ordinary scalar field is recovered (up to a shift / S) in the limit S ! 1 for the following identification of the model’s parameters: r q0m 1 Sq0 p µ~B ≡ ; R ≡ ; R ≡ Sq m : (9) x q2 1 q m 2 0 T. Trzesniewski´ Compact phase space and cosmology 5 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Total Hamiltonian As the result, the matter Hamiltonian acquires the form ! q q π2 1 H = −Nm 0 S = Nq −Sm 0 + ' + m2'2 + O(4) : (10) Smat q x q2 2q2 2 The first term in the expansion will lead to a cosmic bounce, while the negative energy density that occurs for Sx > 0 should be balanced by some additional matter content. Nevertheless, in what follows we will use the positive-definite Hamiltonian q H := Nm 0 (S − S ) (11) mat q x and then the total Hamiltonian is 3κ H = H + H ; H = − Nqp2 ; (12) tot FRW mat FRW 4 @ where κ ≡ 8πG. It generates the constraint @N Htot = 0, equivalent to q 3κ m 0 (S − S ) = p2 : (13) q2 x 4 T. Trzesniewski´ Compact phase space and cosmology 6 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Friedmann equation Introducing the Hubble factor h ≡ q_ =(3q), we now find that the Fried- mann equation is given by (for the gauge N = 1) q κ h2 = m 0 (S − S ) ≡ ρ ; (14) q2 x 3 where ρ denotes the matter energy density. If we express it in terms of the energy density and pressure of an ordinary scalar field π2 1 π2 1 ρ := ' + m2'2 ; P := ' − m2'2 ; (15) ' 2q2 2 ' 2q2 2 we may obtain corrections to the usual Friedmann equation 2 2 κ κ q 2 1 2 2 h = ρ' − ρ' − P' + O(1=S ) : (16) 3 9 Sq0m 2 They become relevant for large q and then can trigger a recollapse. T. Trzesniewski´ Compact phase space and cosmology 7 / 14 Homogeneous cosmology Spherical phase space Perturbative inhomogeneities Dynamics of the model Evolution equations Our Hamiltonian Htot also leads to the following equations for gravity 3κ q_ = − Nqp ; 2 3κ q S _ = 2 + 0 − − y p Np Nm 2 S Sx Sy arctan (17) 4 q Sx and for the field 3κ S _ y Sx = Np Sy arctan ; (18) 2 Sx 3κ S _ y _ Sy = Nm Sz − Np Sx arctan ; Sz = −Nm Sy : (19) 2 Sx π π π π Alternatively, on the hemisphere '=R1 2 (− 2 ; 2 ), π'=R2 2 (− 2 ; 2 ) one may use the angular-like variables, which are governed by NR2 π' ' 2 ' '_ = tan cos ; π_ ' = −NR1qm sin : (20) q R2 R1 R1 T. Trzesniewski´ Compact phase space and cosmology 8 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Hamiltonian for the inhomogeneous field In this case let us restrict to the regime of S ! 1 but with ∆ 6= 0. Then our matter field Hamiltonian becomes (we also choose N = a) Z 3 H' = d x H' " # Z π2 (r')2 1 ∆ = d 3x a4 ' + + m2'2 + (rπ )2 : (21) 2a6 2a2 2 2m2a8 ' 0 Changing the variables to v := a ' and πv := @Lv =@v we can derive π2 (rv)2 1 ∆ H = v + + m2 v 2 + (rπ − h rv)2 v 2 2 2 eff 2m2a2 v + O(∆2) ; (22) 2 2 2 00 0 where meff ≡ m a −a =a is the effective mass, h ≡ a =a the conformal Hubble factor and we make an expansion around ∆ = 0. T. Trzesniewski´ Compact phase space and cosmology 9 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Quantum field operators Since in the considered limit S ! 1 we simply have (3) δ (x − y) (3) f'(x); π'(y)g = −! δ (x − y) ; (23) cos(π'(x)=R2) the standard quantization can be applied, leading to (3) ^ [v^(x); π^v (y)] = iδ (x − y) I : (24) Furthermore, we Fourier expand the field operators Z d 3k Z d 3k v^(x) = eik·xv^ ; π^ (x) = eik·xπ^ (25) (2π)3=2 k v (2π)3=2 vk and decompose their modes in the basis of creation and annihilation y (3) operators, satisfying [a^k; a^q] = δ (k − q),(τ is the conformal time) ^ ^ ∗ ^y vk(τ) = fk (τ) ak + fk (τ) a−k ; (26) ^ ∗ ^y π^vk(τ) = gk (τ) ak + gk (τ) a−k : (27) T. Trzesniewski´ Compact phase space and cosmology 10 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Dynamics of mode functions Next we may write down the (symmetrized) quantum Hamiltonian 1 Z ∆k 2 H^ = O(∆2) + d 3k 1 + π^ π^y +π ^y π^ v 4 m2a2 vk vk vk vk 1 Z ∆k 2 + d 3k !2 + h2 v^ v^y + v^yv^ 4 k m2a2 k k k k 1 Z ∆k 2 − d 3k h v^ π^y + v^yπ^ +π ^ v^y +π ^y v^ ; (28) 4 m2a2 k vk k vk vk k vk k 2 2 2 with !k ≡ k + meff. It determines the evolution equations of v^k and π^vk, which together give us equations for mode functions: 2 2 00 ∆k 0 ∆k f + 2h f + !2 + k 2 + m2a2 − 2h2 f = 0 ; (29) k m2a2 k k m2a2 k as well as 2 0 ∆k 0 g = f + (h f − f ) + O(∆2) : (30) k k m2a2 k k T. Trzesniewski´ Compact phase space and cosmology 11 / 14 Homogeneous cosmology Quantization of the model Perturbative inhomogeneities First order corrections Vacuum state normalization Therefore, the Wronskian condition also becomes modified 2 0 0 ∆k f (f ∗) − f ∗f = i 1 + + O(∆2) : (31) k k k k m2a2 We now calculate that energy of the initial ground state is given by 1 Z h0jH^ j0i = δ(3)(0) d 3k E ; v 2 k ∆k 2 ∆k 2 2∆k 2 E ≡ 1 + jg j2 + !2 + h2 jf j2 + h f g∗ : (32) k m2a2 k k m2a2 k m2a2 k k iα The form of fk can be found by applying the decomposition fk = rk e k 2 2 and looking for a minimum of Ek .
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