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Classical properties of quantum fluctuations in the early universe
Vistasin Astronomy, Vol.37, pp. 507-510,1993 Printedin GreatBritain.Allrights reserved.
0063-6656/93 $24.00 @ 1993 PergamonPress Ltd
CLASSICAL PROPERTIES OF QUANTUM FLUCTUATIONS IN THE EARLY UNIVERSE A. L. Matacz Physics Department, University of Adelaide, G.P.O. Box 498, Adelaide, Australia 5001
INTRODUCTION One of the most attractive features of the Inflationary Universe scenario is its ability to explain the origin of initial energy density inhomogeneities, required to seed galaxies and galaxy clusters (Brandenberger, 1989). During inflation initial quantum fluctuations of the ground state of the inflaton undergo significant amplification after Hubble crossing, leading to a macroscopic quantum state. In much of the previous work on this subject macroscopic was incorrectly taken to be synonymous with classical thus the origin of classical density inhomogeneities was not properly addressed. In actual fact the quantum state of the inflaton is homogeneous. It is the presence of quantum coherence that prevents the appearence of inhomogeneities. Much of this early confusion possibly arose because the usual methods for representing quantum fluctuations do not clearly differentiate the quantum and classical properties of the fluctuations. In this paper I derive the the coherent state representation of quantum fluctuations in an expanding F R W universe. I explicitly solve for the case of massless, minimally coupled fluctuations in a de Sitter phase. This representation shows clearly in what sense the quantum fluctuations are quantum and in what sense they are dassical. More importantly it should be a valuable tool for studying the quantum to classical transition in the early universe.
THE MODEL The action for a free scalar field in the spatially flat expanding metric ds a
= aa(zI)[dr/a- ~ dz~]
(1)
can be written as
(2)
508
A. L
Matacz
where a dot denotes a derivative with respect to conformal time. If we deal with a rescaled field variable0 = a~b, drop a surface term, and expand the scalar field into modes in a box of co-moving volume L s then the model reduces to =
oe
= 1 +-
H0/) = 21~-~0 k -[p~'+
(k' +rn'a' + 6 ( a-'._ a)q~,]
(4)
. [ql;cosk. £ + q~sin~:. ~
(5)
O(x) =
where the sum is over positive k only since each mode is a standing wave rather than a travelling wave. The system is quantised by promoting p~, q~ to operators obeying the usual harmonic oscillator commutation relation. Thus the dynamics is reduced to the dynamics of time-dependent harmonic oscillators. The hamiltonian is not unique since it depends on the choice of canonical variables. Each different hamiltonian selects a different vacuum state for the scalar field. We use this harniltonian since it has most in common with the usual fiat space harmonic oscillator and the vacuum state it selects has several desirable features (Weiss, 1986). The dynamics in the Schr6dinger picture is reduced to solving
(6)
~(,1)0(,7,,1') = i~0(,7,,1')
subject to the initial condition U0/' , r/') = 1. Dropping all mode labels we find that the solution for (6) takes the form
(~)
0(~, ~') = ~(~, ~)k(o)
where
(8) and
a=
k# + ir~ v~'
at = k# - i~ v~
(9)
are the usual creation and annihilation operators. S and R are called squeeze and rotation operators respectively (Schumaker, 1986). The time dependent parameters of these operators are determined through 1 + p + #* + lul 2 (10) tanh 2 r -- 1 - p - p* + I~12 e4i~=
1-p*+#+lp[ 2 1 + p * - p - IPl 2
0 = k(rI - rf) + ~ fn' drl ( mza2 +
-"
(11)
~ (1 " ~ J ' - - ~ * ) )
(12)
where
P = kg and
(13) -
+ (e +
+ C-o -
fi
g -- 0
(1,)
Quantum Fluctuations
509
which is the classical equation of motion. Since all parameters in the propagator must be zero at t / = T/¢the solution of (14) must be so that # = - 1 at t / = t/'. We shall concern ourselves only with initial states that are vacuum states with respect to the annihilation operator. In this case the rotation operator contributes only a time dependent phase factor and the state evolves as a squeezed vacuum It, ~) = e-i°/2S(r, ¢)10). It is also possible to write the squeezed vacuum as Ir, ¢) = e - i e / l ( 2 r s i n h r ) - l / l / ~ : exp I-y2 \(1 2-t atanh_r nhr ~ ] ]] liyeiC~)dY
(15)
where the expansion is over coherent states (Louisell, 1973) defined as eigenstates of the annihilation operator, ftia I = ala I where ct is complex. These gaussian states are minimum uncertainty packets in ~ and ~ with mean values determined by:
= - - ~ ( k ~ + i~)
(16)
Physically each coherent state corresponds to a standing wave inhomogeneity of wavelength 2~rik as can be seen by equation (5). The real part of a determines the amplitude of the wave while the imaginary part determines the canonical momentum of the wave. The formalism we have presented is useful for investigating classical correlations. The notion of classical correlations was first introduced by Halliwell (Halliwell, 1987). He showed that the Wigner function of a mode of the quantum fluctuations was, under special circumstances, peaked about the classically correlated path in phase space. The use of the Wigner function for predicting classical correlations has been criticized (Anderson, 1990) in favour of a coarse-grained Wigner function. I have calculated this coarse grained Wigner function for for the squeezed vacuum (15) and found the peak not to be classically correlated. The representation (15) gives us a more transparent way of understanding classical correlations. Because ~ is a function of time we have written the squeezed vacuum as a superposition over coherant states following trajectories in phase space. In this sense classical correlations would mean the trajectories obey hamiltons equation which however is not the case. Another interesting question is does an initial coherent state propagated by (7) follow a classical trajectory. The answer is no.
QUANTUM FLUCTUATIONS IN A DE SITTER PHASE We shall consider the massless minimally coupled version of (2) in a de Sitter phase where a = -1/HT 1 and (13) becomes --i -- k3~ 3
# - kT/(1 + k2T/2)
(17)
This satisfies the required boundary condition p ~ - 1 as kt/~ ~ - c o . The limit of interest is Ikt/I < < 1 which is long after Hubble crossing. Using (17) and (10-11) we find that, in this limit, r --* co, ~b~ r / 2 and (15) becomes:
I,', ¢) --> ~-,o1~
_ exp (-y~k~,l ~) I - y)dy
(i8)
510
A. L Matacz
With this result the following picture emerges. The quantum fluctuations for a mode k, long after Hubble crossing, take the form of a continous superposition over standing waves of ever increasing amplitude with wavelength 2~/k. The quantum properties of the fluctuations axe clearly evident from the quantum coherence between macroscopic field inhomogeneities which in turn keeps the total state homogeneous. The classical properties are evident because the expansion (18) is over real parameter coherent states. This means that the interfering standing waves are no longer oscillating. This 'freezing' of the quantum fluctuations after Hubble crossing is consitent with classical expectations since classically modes stop oscillating after Hubble crossing.
DISCUSSION A commonly used criteria for classicality of fluctuations is sufficient decoherence in the coordinate representation and peaking of a suitable quantum phase space distribution about a classically correlated trajectory. The role of decohereuce should be to tell us when our quantum phase space distribution has lost its coherence and thus can be treated as a classical probability distribution. Clearly what is required for this is decoherence in phase space and not decoherence in a coordinate representaion. In another work (Laflamme R.L and Matacz A.L, 1992) we have shown the absence of phase space decoherence in a model with considerable coordinate space decoherence. This illistrates the unreliability of using coordinate space decoherence as a criteria for classicality. On the other hand the coherent state representation of the density matrix contains both coherence information and a phase space probability distribution within the one formalism. When a density matrix becomes diagonal in a coherent space representation the probability distribution of the diagonal elements should become our classical phase space probabihty distribution. This paper is the first stage of a study into the quantum to classical transition of fluctuations within the coherent state formalism. The next .step is to introduce a decoherence mechanism. An exactly solvable phenomonlogical model should be white noise linearly coupled the the field and its covariant derivative. I hope to report on this in the future.
REFERENCES Anderson, A. (1990) On predicting correlations from Wigner functions. Phys. Rev. D. 42, 585. Brandenberger, R. H. (1989) Cosmic strings, inflationary universe models and the formation of structure. J. Phys. G: Nucl. Part. Phys. 15, 1. HalliweU, J. J. The quantum to classical transition in inflationary universe models. Phys. Left. B. 196, 444. Laltamme, R,. L. and Matacz, A. L. (1992) Decoherence functional and inhomogeneities in the early universe. In preparation. Louisell, W. H. (1973) Quantum Statistical Properties of Radiation. John Wiley. Schumaker, B. L. (1986) Quantum mechanical pure states with ganssian wavefunetions. Phys. Rep. 135, 317. Weiss, N. (1986) Consistency of hamiltonian diagonalization for field theories in a RobertsonWalker background. Phys. Rev. D. 34, 1768.
0063-6656/93 $24.00 @ 1993 PergamonPress Ltd
CLASSICAL PROPERTIES OF QUANTUM FLUCTUATIONS IN THE EARLY UNIVERSE A. L. Matacz Physics Department, University of Adelaide, G.P.O. Box 498, Adelaide, Australia 5001
INTRODUCTION One of the most attractive features of the Inflationary Universe scenario is its ability to explain the origin of initial energy density inhomogeneities, required to seed galaxies and galaxy clusters (Brandenberger, 1989). During inflation initial quantum fluctuations of the ground state of the inflaton undergo significant amplification after Hubble crossing, leading to a macroscopic quantum state. In much of the previous work on this subject macroscopic was incorrectly taken to be synonymous with classical thus the origin of classical density inhomogeneities was not properly addressed. In actual fact the quantum state of the inflaton is homogeneous. It is the presence of quantum coherence that prevents the appearence of inhomogeneities. Much of this early confusion possibly arose because the usual methods for representing quantum fluctuations do not clearly differentiate the quantum and classical properties of the fluctuations. In this paper I derive the the coherent state representation of quantum fluctuations in an expanding F R W universe. I explicitly solve for the case of massless, minimally coupled fluctuations in a de Sitter phase. This representation shows clearly in what sense the quantum fluctuations are quantum and in what sense they are dassical. More importantly it should be a valuable tool for studying the quantum to classical transition in the early universe.
THE MODEL The action for a free scalar field in the spatially flat expanding metric ds a
= aa(zI)[dr/a- ~ dz~]
(1)
can be written as
(2)
508
A. L
Matacz
where a dot denotes a derivative with respect to conformal time. If we deal with a rescaled field variable0 = a~b, drop a surface term, and expand the scalar field into modes in a box of co-moving volume L s then the model reduces to =
oe
= 1 +-
H0/) = 21~-~0 k -[p~'+
(k' +rn'a' + 6 ( a-'._ a)q~,]
(4)
. [ql;cosk. £ + q~sin~:. ~
(5)
O(x) =
where the sum is over positive k only since each mode is a standing wave rather than a travelling wave. The system is quantised by promoting p~, q~ to operators obeying the usual harmonic oscillator commutation relation. Thus the dynamics is reduced to the dynamics of time-dependent harmonic oscillators. The hamiltonian is not unique since it depends on the choice of canonical variables. Each different hamiltonian selects a different vacuum state for the scalar field. We use this harniltonian since it has most in common with the usual fiat space harmonic oscillator and the vacuum state it selects has several desirable features (Weiss, 1986). The dynamics in the Schr6dinger picture is reduced to solving
(6)
~(,1)0(,7,,1') = i~0(,7,,1')
subject to the initial condition U0/' , r/') = 1. Dropping all mode labels we find that the solution for (6) takes the form
(~)
0(~, ~') = ~(~, ~)k(o)
where
(8) and
a=
k# + ir~ v~'
at = k# - i~ v~
(9)
are the usual creation and annihilation operators. S and R are called squeeze and rotation operators respectively (Schumaker, 1986). The time dependent parameters of these operators are determined through 1 + p + #* + lul 2 (10) tanh 2 r -- 1 - p - p* + I~12 e4i~=
1-p*+#+lp[ 2 1 + p * - p - IPl 2
0 = k(rI - rf) + ~ fn' drl ( mza2 +
-"
(11)
~ (1 " ~ J ' - - ~ * ) )
(12)
where
P = kg and
(13) -
+ (e +
+ C-o -
fi
g -- 0
(1,)
Quantum Fluctuations
509
which is the classical equation of motion. Since all parameters in the propagator must be zero at t / = T/¢the solution of (14) must be so that # = - 1 at t / = t/'. We shall concern ourselves only with initial states that are vacuum states with respect to the annihilation operator. In this case the rotation operator contributes only a time dependent phase factor and the state evolves as a squeezed vacuum It, ~) = e-i°/2S(r, ¢)10). It is also possible to write the squeezed vacuum as Ir, ¢) = e - i e / l ( 2 r s i n h r ) - l / l / ~ : exp I-y2 \(1 2-t atanh_r nhr ~ ] ]] liyeiC~)dY
(15)
where the expansion is over coherent states (Louisell, 1973) defined as eigenstates of the annihilation operator, ftia I = ala I where ct is complex. These gaussian states are minimum uncertainty packets in ~ and ~ with mean values determined by:
= - - ~ ( k ~ + i~)
(16)
Physically each coherent state corresponds to a standing wave inhomogeneity of wavelength 2~rik as can be seen by equation (5). The real part of a determines the amplitude of the wave while the imaginary part determines the canonical momentum of the wave. The formalism we have presented is useful for investigating classical correlations. The notion of classical correlations was first introduced by Halliwell (Halliwell, 1987). He showed that the Wigner function of a mode of the quantum fluctuations was, under special circumstances, peaked about the classically correlated path in phase space. The use of the Wigner function for predicting classical correlations has been criticized (Anderson, 1990) in favour of a coarse-grained Wigner function. I have calculated this coarse grained Wigner function for for the squeezed vacuum (15) and found the peak not to be classically correlated. The representation (15) gives us a more transparent way of understanding classical correlations. Because ~ is a function of time we have written the squeezed vacuum as a superposition over coherant states following trajectories in phase space. In this sense classical correlations would mean the trajectories obey hamiltons equation which however is not the case. Another interesting question is does an initial coherent state propagated by (7) follow a classical trajectory. The answer is no.
QUANTUM FLUCTUATIONS IN A DE SITTER PHASE We shall consider the massless minimally coupled version of (2) in a de Sitter phase where a = -1/HT 1 and (13) becomes --i -- k3~ 3
# - kT/(1 + k2T/2)
(17)
This satisfies the required boundary condition p ~ - 1 as kt/~ ~ - c o . The limit of interest is Ikt/I < < 1 which is long after Hubble crossing. Using (17) and (10-11) we find that, in this limit, r --* co, ~b~ r / 2 and (15) becomes:
I,', ¢) --> ~-,o1~
_ exp (-y~k~,l ~) I - y)dy
(i8)
510
A. L Matacz
With this result the following picture emerges. The quantum fluctuations for a mode k, long after Hubble crossing, take the form of a continous superposition over standing waves of ever increasing amplitude with wavelength 2~/k. The quantum properties of the fluctuations axe clearly evident from the quantum coherence between macroscopic field inhomogeneities which in turn keeps the total state homogeneous. The classical properties are evident because the expansion (18) is over real parameter coherent states. This means that the interfering standing waves are no longer oscillating. This 'freezing' of the quantum fluctuations after Hubble crossing is consitent with classical expectations since classically modes stop oscillating after Hubble crossing.
DISCUSSION A commonly used criteria for classicality of fluctuations is sufficient decoherence in the coordinate representation and peaking of a suitable quantum phase space distribution about a classically correlated trajectory. The role of decohereuce should be to tell us when our quantum phase space distribution has lost its coherence and thus can be treated as a classical probability distribution. Clearly what is required for this is decoherence in phase space and not decoherence in a coordinate representaion. In another work (Laflamme R.L and Matacz A.L, 1992) we have shown the absence of phase space decoherence in a model with considerable coordinate space decoherence. This illistrates the unreliability of using coordinate space decoherence as a criteria for classicality. On the other hand the coherent state representation of the density matrix contains both coherence information and a phase space probability distribution within the one formalism. When a density matrix becomes diagonal in a coherent space representation the probability distribution of the diagonal elements should become our classical phase space probabihty distribution. This paper is the first stage of a study into the quantum to classical transition of fluctuations within the coherent state formalism. The next .step is to introduce a decoherence mechanism. An exactly solvable phenomonlogical model should be white noise linearly coupled the the field and its covariant derivative. I hope to report on this in the future.
REFERENCES Anderson, A. (1990) On predicting correlations from Wigner functions. Phys. Rev. D. 42, 585. Brandenberger, R. H. (1989) Cosmic strings, inflationary universe models and the formation of structure. J. Phys. G: Nucl. Part. Phys. 15, 1. HalliweU, J. J. The quantum to classical transition in inflationary universe models. Phys. Left. B. 196, 444. Laltamme, R,. L. and Matacz, A. L. (1992) Decoherence functional and inhomogeneities in the early universe. In preparation. Louisell, W. H. (1973) Quantum Statistical Properties of Radiation. John Wiley. Schumaker, B. L. (1986) Quantum mechanical pure states with ganssian wavefunetions. Phys. Rep. 135, 317. Weiss, N. (1986) Consistency of hamiltonian diagonalization for field theories in a RobertsonWalker background. Phys. Rev. D. 34, 1768.