State-dependent graviton noise in the equation of geodesic deviation

We consider an equation of the geodesic deviation appearing in the problem of gravitational wave detection in an environment of gravitons. We investigate a state-dependent graviton noise (as discussed in a recent paper of Parikh,Wilczek and Zahariade) from the point of view of the Feynman integral and stochastic differential equations. The evolution of the density matrix in an environment of gravitons is obtained. We express the time evolution by a solution of a stochastic geodesic deviation equation with a noise dependent on the quantum state of the gravitational field.


I. INTRODUCTION
The study of the effect of an environment, which is not directly observable, on the motion of macroscopic bodies, began with the phenomenon of Brownian motion. The mathematical theory of Brownian motion allowed to confirm the presence of invisible molecules ( see [1]). In a recent paper Parikh,Wilczek and Zahariade (PWZ) [2] suggest that in a similar way we can detect the environment of gravitons in spite of the negative conclusions of Dyson [3]. According to PWZ the noise resulting from the ground state or coherent state of gravitons is weak and practically undetectable. However, the high temperature states and squeezed states of the graviton can lead to a substantial increase of noise which can be detected in the gravitational wave detection experiments. The authors [2] derive their results by an investigation of the geodesic deviation equation in the environment of the quantized gravitational field. They apply the influence functional method [4] in order to transform the evolution of the transition probability of macroscopic bodies into an expectation value with respect to the noise disturbing the geodesic deviation equation. Studies on the effect of gravitons on the motion of other particles appeared earlier [5][6] [7][8] [9] [10], but these papers did not tackle directly the geodesic deviation equation. The quantum noise from the environment has been investigated in other fields of physics [11] [12]. It can be detected in quantum optics [13] and solid state physics.
The evolution of the density matrix in an environment of unobservable particles is usually treated by means of the influence functional [4] which starts from the Feynman integral. As a result one can express the evolution of the density matrix as a solution of the master equation. One can also express the density matrix by the Wigner function and derive a stochastic equation for the evolution of the Wigner function. We have derived such equations for the thermal graviton environment by means of the Feynman integral in [7]. In this paper we apply the method to the quantum evolution of a particle following the geodesic deviation equation when the quantum gravi- * Electronic address: zbigniew.haba@uwr.edu.pl tational field is in an arbitrary Gaussian state. We apply a method of transforming the Feynman integral into an expectation value with respect to the Brownian motion developed in [14] [15]. In such a case the dependence of the evolution of the density matrix on the state of the gravitational field is exhibited explicitly.
The plan of the paper is the following. In sec.2 we derive in a novel way the transformation of the Feynman integral in quantum mechanics into an expectation value over a state-dependent noise. In sec.3 we show how this transformation applies to quantum field theory. In sec.4 we discuss in detail perturbations by noise resulting from the time-dependent Gaussian states. In sec.5 we apply the method to the BPZ model (as introduced in [2]) in the one mode approximation . In sec.6 the influence of infinite number of modes of the thermal gravitational field upon the geodesic deviation equation is treated by the method of the forward-backward Feynman integral as previously applied in [7] to a geodesic motion. In sec.7 we apply our method of the state-dependent noise to derive a stochastic deviation equation which governs the evolution of the density matrix. Sec.8 contains a summary of the results. In the Appendix we discuss the stochastic deviation equation which results from the assumption that the initial values of the classical gravitational field have the thermal Gibbs distribution.

II. STATE-DEPENDENT TRANSFORMATION OF THE FEYNMAN INTEGRAL
Let us consider first a simple model of the Schrödinger equation of quantum mechanics in one dimension with the time-dependent potential V t The solution ψ t with the initial (boundary) condition ψ 0 at t = 0 can be expressed by the Feynman integral where q 0 (x) = x.
Assume that the initial condition is a product of the wave functions We define a stochastic process q s depending on ψ g 0 and defined as a functional of the Brownian motion b s . The stochastic process q s is defined as a solution of the stochastic equation Here, the Brownian motion is defined as the Gaussian process with the covariance We show that with a proper choice of α and a the solution of the Schrödinger equation (1) can be expressed as where q t (x) is the solution of eq.(4) with the initial con- The idea is that if in eq.(2) ψ 0 (q t ) = ψ g 0 (q t )χ 0 (q t ) then we can represent ψ g 0 (q t ) = exp ln ψ g 0 (q t ) as an integral With a proper choice of α and a in eq.(4) the potential term in eq.(2) will cancel. More precisely we shall have So that the integration over q in the Feynman formula (1) will be replaced by an average over Brownian paths b t . It is crucial for the derivation that functionals of the Brownian motion satisfy a modified differential formula (Ito formula [16]) following from the non-differentiability of the Brownian paths b t where • denotes the Stratonovitch differential (the differential db without circle is called Ito differential [16]) and on the rhs of (9) we insert < (db) 2 >= ds (we denote E[..] by < .. > as a shorthand). If the stochastic equation (4) is satisfied and if f s is a function of q We insert f s = ln ψ g t−s .Then d ln ψ g t−s (q s ) = ∇ ln ψ g t−s dq s + 1 2 a 2 ∇ 2 ln ψ g t−s ds+∂ s ψ g t−s ds.
We have We use the fact that ψ g t−s satisfies the Schrödinger equation (1). Then, in equation (10) we can express ∇ 2 ψ t−s by V t−s and ∂ s ψ g t−s . We can check after an insertion of eqs. (7) and (10) We can repeat the procedure for the Feynman integral solving Schrödinger equation with a final boundary where q t (x, t) = x. Let ψ τ = ψ g τ χ τ where ψ g τ is the solution of eq.(1). As before we use the identity We perform the differentiation d ln ψ as in eq.(10). We check that if q s satisfies the equation then the Feynman integration is reduced to the Brownian average The imaginary time version of our formulas (at least the ones when ψ g is the ground state) belongs to the standard theory of stochastic differential equations (see [17] for eq.(11) and [18] for eq. (14)). The derivation of the Feynman integral has been discussed earlier in a different form in [14] [15]. There are some mathematical subtleties concerning analytic properties of functions of the stochastic processes as our formalism is an analytic continuation of the one well known for the imaginary time.
The formalism can be extended to arbitrary number of dimensions and to Lagrangians of the form Then the Schrödinger equation (1) holds with the Hamiltonian where g = det[g jk ] , g jk are the matrix elements of the inverse matrix and is the momentum in classical mechanics . In quantum mechanics The corresponding stochastic equation (an analog of eq.(11)) reads where e j l e k l = g jk . As a simple example we could take the harmonic oscillator with the ground state solution The stochastic equation (11) reads A simple calculation gives (23) The rhs of eq.(23) is the expectation value in the ground state of the time-ordered products of Heisenberg picture position operators of the harmonic oscillator. In this sense the stochastic process q t has a physical meaning as its correlation functions coincide with quantum expectation values.

III. THE RELATIVISTIC QUANTUM FIELD THEORY
In the canonical field theory with the Hamiltonian where consider the Schrödinger equation and its imaginary time version Let where ψ g is the ground state. Then, χ satisfies the equationh where Eq.(28) is a diffusion equation in infinite dimensional spaces. An approach to Euclidean field theory based on this equation has been developed in [19]. It follows that the solution of eq.(28) can be expressed as where Φ t (Φ) is the solution of the stochastic equation with the initial condition Φ. E[...] denotes an expectation value with respect to the Wiener process (Brownian motion) defined by the covariance The correlation functions of the Euclidean field in the ground state ψ g can be expressed by the correlation functions of the stochastic process Φ t . Let us consider the simplest example:the free field. Then, the ground state is Eq.(31) reads with the solution (with the initial condition Φ at t 0 ) We can calculate In eq.(36) the rhs denotes the kernel of the operator.
If the ground state has some analycity property then we can extend the time in eqs.(25)-(36) t → it from imaginary time to the real time [14] [15]. An analytic continuation to real time of eq.(34) reads The solution is (38) The solution of the Schrödinger equation (25) has the same form (6) with Φ t (Φ) of eq.(38) replacing q t (x). We can see that is the quantum field and T denotes the time-ordered product.
We can transform the complex equation (37) into two real equations defining and Then it follows from eq.(37) that and φ − satisfies the random wave equation which has the solution where ∂ t φ − (0) = ωφ + (0).

IV. TIME-DEPENDENT REFERENCE STATE
There are time-dependent solutions in the Gaussian form of the Schrödinger equation for the free field theory In the time-dependent case eq.(11) in quantum field theory takes the form [17] Let Φ s (Φ) be the solution of eq.(47) with the initial condition Φ then the solution of the Schrödinger equation is With the time-dependent interaction V t in eq.(24) the Feynman formula reads (where ψ g t is the solution of free field theory) If we impose the final boundary condition at t = τ on the solutions Φ t (Φ, τ ) of the stochastic equation then we can express the solution of the Schrödinger equation with the final condition χ τ at t = τ as [18] ψ Let us decompose eq.(50) (J = 0)into the real and imaginary parts Then Expressing φ − by φ + from eq.(54) and inserting it in eq.(55) we obtain an equation Hence, φ + satisfies the wave equation with friction (we further develop the theory of stochastic wave equations for the quantum theory of fields in expanding universes in [21] as an extension of Starobinsky stochastic inflation [20]).
As an application to the particle motion in a (quantized) gravitational wave let us consider the Hamiltonian of a harmonic oscillator Let us look for a time-dependent solution of the Schrödinger equation in the form Then, ψ g is a solution of the Schrödinger equation (1) if and It can be seen that the Riccatti equation (60) is equivalent to where Then, J = Cu −1 with a certain constant C . The general solution of eq.(61) is where σ, δ are complex numbers. Eq.(47) (J = 0) reads dq s = −∂ t ln u t−s q s ds + √ ihdb s = −ω(−σ sin(ω(t − s)) + δ cos(ω(t − s))) (σ cos(ω(t − s)) + δ sin(ω(t − s))) −1 q s ds + √ ihdb s (64) with the solution We have If δ σ = ia then Γ(0) = iaω and the solution ψ t starts from a real Gaussian function. The case δ = iσ corresponds to the ground state (21) (m = 1). It can be shown that the formula (66) is continuous with respect to the limit δ → iσ.

V. PARTICLE INTERACTING WITH GRAVITONS:PWZ MODEL IN ONE MODE APPROXIMATION
Parikh,Wilczek and Zahariade [2] consider two masses M and m 0 interacting with gravitational field q ω . In a one mode approximation the Lagrangian describing the geodesic deviation of the m 0 mass ( in the free falling frame) is [2][22] where λ = √ 8πG, x = (q ω , ξ), ξ is the geodesic deviation.
The Lagrange equation is The Hamiltonian is determined by eq. (17) with We could quantize the model (67) with the explicit quantum version (17) of the Hamiltonian. However, the averaging over gravitons is simple only in the influence functional approach. In general, the wave function evolution in a potential V and in the gravitational field g µν is where L(g) is the gravitational Lagrangian and q are the initial values of the metric. The density matrix in an environment of gravitons described by variables q is In the PWZ model (67), where the gravitational field is described by one mode q ω , when we apply the transformation (8) to q with the ground state (21) (m = 1) then from eq.(49) we obtain where q * is another realization of the process q (and the complex conjugation of this realization), we have denoted the evolution kernel of the density matrix by K t .
The calculation of the expectation value (72) gives (we use the solution of eq. (22) and assume that the initial condition χ 0 does not depend on q ) In eq.(73) we have (from eq. (22)) (74) If the oscillator is in a time-dependent state then we should insert the solution (63) in the Feynman formula (72). Hence, instead of eq.(73) we have After calculation of the q integral in eqs. (73) and (75) we obtain a quadratic functional of f s and f ′ s ′ in the exponential. In the simplest case (73) of the ground state of the oscillator we obtain We expand the exponential in eq.(77) in y. Then, the terms independent of y cancel and there remains (up to the terms quadratic in y ) (79) The term linear in y gives a modification of the equation of motion of the ξ coordinate whereas the term quadratic in y is a noise acting upon the particle [7].
In the expression (75) of the time-dependent reference state we obtain where by L we denoted a functional of X, M is an operator and byL we denoted the term proportional to y. If we introduce Y = M − 1 2L then Y becomes a Gaussian variable with the white noise distribution which can be represented as ∂ s W . The factor depending on y is integrated out contributing just a constant. The calculation of ρ t is reduced to an average over solutions of the stochastic equation In the next section dealing with a thermal state of the gravitational field we approximate L and M by local functions of X.

VI. INFINITE NUMBER OF MODES:THERMAL STATE OF GRAVITONS
We are to generalize the results of sec.5 to an infinite number of modes of the gravitational field. We have studied earlier [7] an analogous model of a particle geodesic motion in an environment of quantum gravitational waves in a thermal state. There are minor changes from the setting of Parikh, Wilczek and Zahariade [2]where the effect of gravitons on geodesic deviation is considered. With the infinite number of gravitational wave modes in the model (67) of sec.5 we shall have ω = |k| (we set the velocity of light c = 1), and we sum up the terms with q ω over k with the measure dk. Now the Lagrangian (67) is [22] where the metric perturbation h rl of the Minkowski metric η µν (g µν = η µν + h µν ) is in the transverse-traceless gauge, k is the wave vector and the geodesic deviation ξ in a gravitational transverse-traceless gauge has only the spatial components ξ r . We apply the decomposition of h rl = h w rl +h q rl into the classical wave solution h w rl and the quantized (graviton ) part h q rl . h q rl is decomposed in the amplitudes h α (where α = +, ×,in the linear polarization ) by means of the polarization tensors e α rl [23][24] The h q rl are quantized at finite temperature T with the Gibbs distribution exp(−βH), β −1 = k B T where k B is the Boltzman constant. The classical part of the particlewave interaction can be considered as a time-dependent external potential V (ξ) with Denote Then, as in [7] (there are some misprints of signs in [7]; for the derivation of the thermal formula see [25],sec.18, see also [5][8] [10];the gravitational case is analogous to the electromagnetic one treated in [26]) we obtain for the density matrix evolution kernel (86) In (86) we have a decomposition of the finite temperature transverse-traceless graviton propagator D into the real and imaginary parts D = A + iC where . We neglect the dependence on x in eqs.(87)-(88). Then the angular average over kk −1 gives When we neglect the x dependence of the propagators (assuming ξ is small in comparison with the gravitational wave length) and average over the angles then the kintegral dk ≃ 4πdkk 2 in the high temperature limit of A rl;mn in eq.(87) gives δ(s − s ′ ). In C rl;mn (88) we write (as in [7]) sin(k(s − s ′ )) = −k −1 ∂ s cos(k(s − s ′ )). Then, integrating over k we obtain ∂ s δ(s − s ′ ) . In such a case the evolution kernel of the density matrix reads where q rl = 1 2 results from a resummation of f rl with < Λ ij;rl > γ = 8πGm 2 0 10π (92) We expand eq.(90) around X (78) (now with three spatial indices) In the exponential (90) the term linear in y becomes The term quadratic in y is the noise term. The calculation of the evolution kernel is reduced to an expectation value over the solutions of the stochastic equation As explained in the derivation of eq.(81) the term quadratic in y defines the operator M . From eq.(90) we obtain that M is an operator defined by the bilinear form (on the rhs of eq.(95) we have the square root of this matrix) y r M rl y l = 2 ds d 2 ds 2 (X j y l ) d 2 ds 2 (X j y l ) We can see that the noise on the rhs of eq.(95) has the factor √ T which can be large at high temperature.

VII. GENERAL GAUSSIAN STATE OF THE GRAVITON
We generalize the results of sec.5 on averaging over the oscillator modes to infinite number of modes of the gravitational field. The gravitational perturbation in the transverse-traceless gauge is decomposed into the amplitudes h α by means of the polarization tensors e α rl ,α = +, ×, h rl = e α rl h α . The gravitational Hamiltonian H = H + + H × has the same form as for two independent scalar fields Φ → h α (eq.(24) with m = V = 0). In such a case we have two independent equations (60) for Γ α and two independent u α . The quantum interaction in eq.(67) in multimode case takes the form where f α = e α rl f rl . We consider a solution of the Schrödinger equation (25) in the Gaussian form (if we did not split the gravitational field into h w and h q then the classical part would be obtained from the term Jh in eq.(57)) In eq.(97) (Γ α − Γ * α ) −1 has a physical meaning as the squeezing of the amplitude h α in the uncertainty relations (squeezed states are produced during inflation [28] [29]). In general, Γ + and Γ × are independent solutions of eq.(60). We restrict our discussion to the case Γ + = Γ × ≡ Γ ( and u + = u × ). Then, the gravitational field resulting from the state (97)  which after averaging over kk −1 will be expressed by q rl (91) in the way analogous to the thermal case of sec.6 (Λ mn;rl f mn s f rl s ′ →< Λ mn;rl > f mn s f rl s ′ ). In detail (99) In eq.(99) the f α f α term has been expressed by q rl and u(k) is defined in eq.(63) with ω = k.
For the final result the integral over k is crucial. In the thermal case the effective action at high temperature in the exponential (90)was local in time (for a small β ) owing to the k −1 factor coming from coth( 1 2 βk). We do not have such a factor here. Nevertheless, we can see that the non-local final noise can be large owing to the squeezing factor (Γ − Γ * ) −1 in eq.(99).Explicitly, this term is (100) The exponential in (99) is of the form When δ σ = i and Γ = ik then in the integral (99) we obtain (this is an infinite mode version of eq.(77)) K t ≃ exp − λ 2 m 2 0 2h 4π 5 dkk t 0 dsds ′ exp(−ik|s − s ′ |)q rl (s)q rl (s ′ ) + exp(ik|s − s ′ |)q ′ rl (s)q ′ rl (s ′ )+ − exp(−ik(s + s ′ ))(q rl (s))q ′ rl (s ′ ) + q ′ rl (s)q rl (s ′ ) .
(102) Representing sin(kt) as −k −1 ∂ t cos(kt) we integrate over k obtaining ∂ s δ(s − s ′ ) in a similar way as in the thermal case arriving at the phase factor We obtain the same lhs of the stochastic equation as in the thermal state , but the noise resulting from eq.(102)is different than the one of eq.(95) ( non-local in time and non-Markovian). In eq.(99) the exponential of the evolution kernel for the density matrix is of the form +V (ξ) − V (ξ ′ )) + i dsds ′ (q rl − q ′rl )C(q rl + q ′rl ) +(q rl + q ′rl )A(q rl + q ′rl ) + (q rl − q ′rl )B(q rl − q ′rl ) .
If the exponential (99)-(101) is written in the form (104) then α 1 = A+B +C, α 2 = A+B −C, α 3 = 2(A−B −C). The functions A, B, C can be read from eqs.(99)-(100). Expanding in y we obtain the non-Markovian stochastic equation as derived in eq.(81). In eq.(104) the Cterm is proportional to y whereas the A and B terms are quadratic in y. So the C term gives the modification of the equation for the geodesic deviation whereas the A, B terms contribute to the noise.

VIII. SUMMARY
We have calculated the average of the density matrix over the gravitational field in a thermal state and in a general Gaussian state. The result shows that for gravitons in high temperature or in a highly squeezed state the modification of the geodesic deviation equation applied in a gravitational wave detection can have large noise amplitude. In general, the perturbation of the geodesic deviation equation is non-local and non-Markovian. Our results on the stochastic geodesic deviation equation show some differences in comparison to PWZ [2] which may come from approximations used by those authors. The formula for the noise (although quite complicated) can be useful in order to distinguish the contribution of the graviton noise from other sources of noise in the gravitational wave detection. The noise could be detectable if gravitational waves come from an inflationary stage of universe evolution (squeezing) or from a merge of hot neutron stars.
where we assumed that at t 0 the first and the second derivatives of f mn are zero. If we assme that in (114) f mn (t 0 ) = 0 then we can write eq.(108) in the form 5π (δ rm δ ln − 1 3 δ rl δ mn )ξ l ∂ 3 t f mn + N rl (t)ξ l .