:THE COSMOLOGICAL OTOC: Formulating New Cosmological Micro-Canonical Correlation Functions for Random Chaotic Fluctuations in Out-Of-Equilibrium Quantum Statistical Field Theory
Abstract
:Contents | ||||
1 | Introduction | 4 | ||
2 | Formulation of OTOC in Cosmology | 18 | ||
2.1 | General Remarks on OTOC……………………………………………………………………………………………… | 18 | ||
2.2 | Eigenstate Representation of OTOC in Quantum Statistical Mechanics…………………………………………… | 20 | ||
2.3 | Constructing OTOC in Cosmology……………………………………………………………………………………… | 23 | ||
2.3.1 | For Massless Scalar Field……………………………………………………………………………………… | 29 | ||
2.3.2 | For Partially Massless Scalar Field……………………………………………………………………………… | 33 | ||
2.3.3 | For Massive Scalar Field……………………………………………………………………………………… | 34 | ||
3 | Quantum Micro-Canonical OTO Amplitudes and OTOC in Cosmology | 36 | ||
3.1 | Computational Strategy……………………………………………………………………………………………… | 37 | ||
3.2 | Classical Mode Functions in Cosmology……………………………………………………………………………… | 38 | ||
3.3 | Quantum Mode Function in Cosmology……………………………………………………………………………… | 41 | ||
3.4 | Canonical Quantization of Cosmological Hamiltonian: Classical to Quantum Map……… | 43 | ||
3.5 | Cosmological Two-Point and Four-Point “In-In" OTO Micro-Canonical Amplitudes……… | 44 | ||
3.5.1 | OTOC Meets Cosmology……………………………………………………………………………………… | 44 | ||
3.5.2 | Fourier Space Representation of the Commutator Bracket: Application to Two-Point OTOC……… | 46 | ||
3.5.3 | Fourier Space Representation of Square of the Commutator Bracket Application to Four-PointOTOC……………………………………………………………………………………………… | 48 | ||
3.6 | Cosmological Thermal Partition Function: Quantum Version……………………………………………………… | 52 | ||
3.6.1 | Quantum Vacuum State in Cosmology……………………………………………………………………… | 52 | ||
3.6.2 | Quantum Partition Function in Terms of Rescaled Field Variable……………………………………… | 52 | ||
3.6.3 | Quantum Partition Function in Terms of Curvature Perturbation Field Variable……………………………………… | 53 | ||
3.7 | Trace of Two-Point “In-In" OTO Amplitude for Cosmology……………………………………………………………………… | 54 | ||
3.8 | OTOC from Regularised Two-Point “In-In" OTO Micro-Canonical Amplitude: Rescaled Field Version……………………………………………………………………………………… | 55 | ||
3.9 | OTOC from Regularised Four-Point “In-In" OTO Micro-Canonical Amplitude: Curvature Perturbation Field Version……………………………………………………………………………………………………………… | 56 | ||
3.10 | Trace of Four-Point “In-In" OTO Amplitude for Cosmology……………………………………………………………… | 58 | ||
3.11 | OTOC from Regularised Four-Point “In-In" OTO Micro-Canonical Amplitude: Rescaled Field Version……………………………………………………………… | 60 | ||
3.11.1 | Without Normalization……………………………………………………………… | 60 | ||
3.11.2 | With Normalization……………………………………………………………… | 64 | ||
3.12 | OTOC from Regularised Four-Point “In-In" OTO Micro-Canonical Amplitude: Curvature Perturbation Field Version……………… | 64 | ||
3.12.1 | Without Normalization…………………………………………………………………………… | 64 | ||
3.12.2 | With Normalization…………………………………………………………………………… | 64 | ||
4 | Numerical ResultsI: Interpretation of Two-Point Micro-Canonical OTOC in Cosmology | 65 | ||
5 | Numerical ResultsII: Interpretation of Four-Point Micro-Canonical OTOC in Cosmology | 73 | ||
6 | Lyapunov Spectrum and Quantum Chaos in Cosmology | 86 | ||
7 | Numerical ResultsIII: Interpretation of Cosmological Lyapunov Spectrum | 89 | ||
8 | Classical Limit of Micro-Canonical OTO Amplitudes and Related OTOC in Cosmology | 93 | ||
8.1 | Computational Strategy……………………………………………………………………………… | 93 | ||
8.2 | Classical Limit of Cosmological Two-Point “In-In" OTO Micro-Canonical Amplitudes……………… | 94 | ||
8.3 | Classical Limit of Cosmological Four-Point “In-In" OTO Micro-Canonical Amplitudes……………… | 95 | ||
8.4 | Cosmological Micro-Canonical Partition Function: Classical Version……………………………………………… | 98 | ||
8.4.1 | Classical Micro-Canonical Partition Function in Terms of Rescaled Field Variable | 98 | ||
8.4.2 | Classical Micro-Canonical Partition Function in Terms of Curvature Perturbation Field Variable…………………………………… | 99 | ||
8.5 | Classical Limit of Cosmological Two-Point Micro-Canonical OTOC: Rescaled Field Version | 99 | ||
8.6 | Classical Limit of Cosmological Two-Point Micro-Canonical OTOC: Curvature Perturbation Field Version………………………………… | 101 | ||
8.7 | Classical Limit of Cosmological Four-Point Micro-Canonical OTOC: Rescaled Field Version……………………………………… | 101 | ||
8.7.1 | Without Normalization……………………………………………………………………………… | 101 | ||
8.7.2 | With Normalization……………………………………………………………………………… | 103 | ||
8.8 | Classical Limit of Cosmological Four-Point Micro-Canonical OTOC: Curvature Perturbation Field Version……………………………………… | 104 | ||
8.8.1 | Without Normalization……………………………………………………………………… | 104 | ||
8.8.2 | With Normalization……………………………………………………………………………… | 104 | ||
9 | Summary and Outlook | 105 | ||
A | Asymptotic Behaviour of the Cosmological Mode Functions | 108 | ||
B | WKB Solution of the Cosmological Mode Functions for Time Dependent Protocols | 115 | ||
C | Quantum Two-Point OTO Micro-Canonical Amplitudes for Cosmology | 116 | ||
C.1 | Definition of Micro-Canonical OTO Amplitude ……………………………… | 116 | ||
C.2 | Definition of Micro-Canonical OTO Amplitude ……………………………… | 117 | ||
D | Quantum Four-Point OTO Micro-Canonical Amplitudes for Cosmology | 117 | ||
D.1 | Definition of Micro-Canonical OTO Amplitude ……………………… | 117 | ||
D.2 | Definition of Micro-Canonical OTO Amplitude ……………………… | 118 | ||
D.3 | Definition of Micro-Canonical OTO Amplitude ……………………… | 119 | ||
D.4 | Definition of Micro-Canonical OTO Amplitude ……………………… | 120 | ||
E | Computation of Classical Limit of Four-Point “In-In” OTO Micro-Canonical Amplitudes for Cosmology | 121 | ||
F | Computation of Quantum Micro-Canonical Partition Function in Cosmology | 124 | ||
F.1 | Quantum Micro-Canonical Partition Function in Terms of Rescaled Field Variable……… | 124 | ||
F.2 | Quantum Micro-Canonical Partition Function in Terms of Curvature Perturbation Field Variable……………………………………… | 125 | ||
G | Computation of Classical Micro-Canonical Partition Function in Cosmology | 126 | ||
G.1 | Classical Micro-Canonical Partition Function in Terms of Rescaled Field Variable……… | 126 | ||
G.2 | Classical Micro-Canonical Partition Function in Terms of Curvature Perturbation Field Variable………………………………………… | 127 | ||
H | Computation of the Trace of the Two-Point Amplitude in Micro-Canonical OTOC | 128 | ||
I | Computation of the Trace of the Four-Point Amplitude in Micro-Canonical OTOC | 130 | ||
J | Time Dependent Two-Point Amplitude in Micro-Canonical OTOC | 135 | ||
K | Time Dependent Four-Point Amplitudes in Micro-Canonical OTOC | 137 | ||
K.1 | Computation of ……………………………………………………………………………………………… | 137 | ||
K.2 | Computation of ……………………………………………………………………………………………… | 151 | ||
K.3 | Computation of ……………………………………………………………………………………………… | 151 | ||
K.4 | Computation of ……………………………………………………………………………………………… | 151 | ||
K.5 | Computation of ……………………………………………………………………………………………… | 152 | ||
K.6 | Computation of ……………………………………………………………………………………………… | 152 | ||
K.7 | Computation of ……………………………………………………………………………………………… | 152 | ||
K.8 | Computation of ……………………………………………………………………………………………… | 153 | ||
K.9 | Computation of ……………………………………………………………………………………………… | 153 | ||
L | Computation of the Normalization Factor in Four-Point Micro-Canonical OTOCC | 153 | ||
L.1 | Normalization Factor of Four-Point Micro-Canonical OTOC Computed from Rescaled Field Variable……………………………………………………………………………………………………… | 153 | ||
L.2 | Normalization Factor of Four-Point Micro-Canonical OTOC Computed from Curvature Perturbation Field Variable……………………………………………………………………………………… | 156 | ||
M | Computation of the Normalization Factor in Classical Limit of Four-Point Micro-Canonical OTOC | 157 | ||
M.1 | Normalization Factor of the Classical Version of Four-Point Micro-Canonical OTOC Computed from Rescaled Field Variable………… | 157 | ||
M.2 | Normalization Factor of the Classical Version of Four-Point Micro-Canonical OTOC Computed from Curvature Perturbation Field Variable…………… | 158 | ||
N | Thermal Trace Operation in Terms ofWave Function of the Universe in Cosmology | 159 | ||
References | 160 |
1. Introduction
- Possibility I:The first possibility pointing towards the future observational aspects which can be probed by different ongoing and upcoming experiments to verify various theoretical features of primordial Cosmology. The most significant quantity which is possible to understand the underlying quantum field theory origin of the primordial physics is the tensor-to-scalar ratio. Detection of this with high statistical accuracy will provide us with the information regarding the generation of primordial gravitational waves, which will further confirm the exact origin of the primordial quantum mechanical fluctuations by exactly estimating the scale of inflation. Not only discriminating different frameworks of inflationary paradigm, but also the existence of alternatives to inflationary frameworks, i.e., bouncing cosmology [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90], cyclic cosmology [91,92,93,94,95,96], ekpyrotic scenario [97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112], etc., can also be verified by the confirmation of the primordial gravity waves. The next important quantity within the framework of primordial Cosmology is the study the existence of non-Gaussian features in the quantum mechanical fluctuations and the probability distribution profile of its related correlation functions. For this purpose, the study of bispectrum and trispectrum, which are basically representing the momentum-dependent amplitude of the three-point and four-point correlation function are very important. In the near future, through the upcoming cosmological missions, if it is possible to detect these non-Gaussian amplitudes with high statistical accuracy then more stringent constraints on the primordial physics are put. This is because of the fact that the probability distribution of such quantum fluctuations in the primordial universe is almost following Gaussian profile and a small but significant deviation from such Gaussianity will be extremely helpful to discriminate amongst various possible theoretical models of inflation which can be able to generate a significant amount of non-Gaussian amplitude in the context of three-point and four-point correlation functions. We are very hopeful for the detection of these important observables in near-future cosmological experiments. For this purpose, one needs to upgrade the present experimental tools and techniques or have to wait for upcoming future advanced experiments, which can able to measure these mentioned observables with high statistical accuracy. Now, we all know that for a single field inflationary paradigm the amplitude of the three point function is directly related to scalar spectral index, given by [38] (in ref. [38], the author have considered an additional contribution which is given by the following expression:Here, it is important to note that is the momentum scale on which the CMB observation is performed and is computed in terms of scalar spectral index using the previously mentioned Maldacena’s consistency relation. All these three relations can be further written in terms of the slow-roll parameters, which can be constructed from the field derivative of the inflationary potentials or the time derivative from the Hubble parameter. Now as we have mentioned, if we can verify this relation through cosmological observation with high statistical accuracy, then not only the appearance of non-Gaussian fluctuations in the primordial cosmological perturbation can be immediately tested, but also the appearance of the scale dependence of primordial power spectrum through the running and running of the scalar spectral index can also be verified by considering the above mentioned consistency relations.
- Possibility II:The second possibility is pointing towards the probing of new physical concepts by incorporating additional but significant features within the framework of early universe Cosmology. Until now all of the features are studied by considering the fact that the quantum fluctuations appearing in the primordial universe is at thermal equilibrium. However, if the quantum mechanical systems which we use to study to describe the early universe Cosmology are not in thermal equilibrium then quantifying the quantum correlation functions within the framework of finite temperature out-of-equilibrium quantum field theory of Cosmology is extremely difficult to compute and until now in the literature of Cosmology no such framework is available using which one can pursue this specific computation. On the other hand, anybody can ask us here why at all computation and quantification of all such quantum mechanical correlation function for the primordial Cosmology describing an out-of-equilibrium is important at all and if these ideas can be provided theoretically then in which stages of the evolutionary scale of our universe this can be really implemented? The answer is very simple and it is already hidden there in the study of the early universe Cosmology. In the following we now explicitly mention about these phenomena where this methodology can be applicable:
- Stochasticity and particle production during inflation:During the epoch of inflation [40,49,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142], we describe the physics of inflationary paradigms with scalar field which have a negligible mass compared to the Hubble scale, which is treated as the reference characteristic scale of early universe Cosmology. However, during inflation many particle produces in a stochastic manner which have masses either of the order of Hubble scale or have mass very heavier than the Hubble scale. These particles are commonly known as partially massless or heavy scalar fields whose origin can be explained from the randomness appearing in the quantum mechanical stochastic fluctuation appearing in the early universe where various unknown out-of-equilibrium features play significant role. However, it is extremely difficult to quantify or describing this phenomenon correctly at the out-of-equilibrium regime of quantum field theory which consistently describe a quantum mechanical theory of primordial Cosmology. Additionally it is important to note that during the stochastic particle production during inflation due to the presence of randomness noise plays significant role to describe the quantum mechanical phenomenon during this particular epoch in the cosmological evolutionary time scale. However, we have a very small understanding regarding the time-dependent random noise within the framework of out-of-equilibrium quantum field theory until now. Only in some specific situations if we provide some additional constraints on the two-point and the one-point correlation of the time-dependent noise, which have the Gaussian probability distribution profile one can deal with noise. However, such analysis within the framework of quantum field theory is not exactly correct which in turn could not able to gives us the correct predictions from the quantum theory of early universe. One the other hand, we really do not know about the nature of the time-dependent noise which we are studying in the present context. This means we do not know that whether the time-dependent noise have Gaussian or non-Gaussian probability distribution profile, which information can be directly understandable if we really know the exact computation of N-point correlation functions of time-dependent noise in cosmology. A lot of attempts have been made within the framework of quantum-field theory to study the quantum mechanical features and the related N-point correlation functions to study the exact nature, behaviour and magnitude of the noise. However, apart from its rigor, not much is known about the detailed quantum filed theory structure from which one can reliably compute such correlation functions as quantum mechanics. Here lies one of the prime motivations to write this paper. Our expectation from the presented computation of this paper is that the cosmological version of OTOC is defined in a specific quantum mechanical vacuum state, which actually describes the initial condition in early universe Cosmology, which describes the randomness and stochastic features of quantum mechanical fluctuations in the out-of-equilibrium regime in a perfectly correct fashion. This is just not an arbitrary claim, but also one can appear at such conclusion by considering the basic understandings of the background physical phenomenon which describes the particle production mechanism during the epoch of inflation. Throughout our paper, we have explicitly established the framework for the first time in the Cosmology literature using which one can explicitly perform the computation of the cosmological correlation functions of the random quantum mechanical fluctuations in the out-of-equilibrium regime of quantum field theory. Instead of studying the quantum correlations with the time ordered or the anti-time ordered physics, in the present context cosmological OTOC functions are playing significant role to describe the underlying hidden features of out-of-equilibrium physics to describe the stochastic randomness during the particle production mechanism during inflation (here it is important to note that the Random Matrix Theory is an another alternative framework using which one can compute these quantum mechanical cosmological correlation functions within the framework of out-of-equilibrium quantum field theory [143,144]).
- Reheating / Warm Inflation:Reheating is an epoch in the evolutionary time scale of our universe which appears just after inflation. During inflation the inflaton field started slowly rolling through the valley of the inflationary potential under consideration and it is expected that after a certain time the inflaton field will reach the stable minimum of the potential and the inflationary mechanism just stops there. However, is the the end of the story? Obviously it is not the end. Once the inflation ends, the inflation field started oscillating around the stable minimum of the potential and started interacting with the valley of the potential as well as some other field, which we identify as the reheaton, the field who is mostly responsible for reheating (it is important to note that in the Cosmology literature there exists a few models of inflation where reheating is not required and without having any reheating phase in those models one can able to describe the genesis of the dark matter candidate theoretically. However, these models are very small in number and the construction of these models relies on various underlying assumptions, which may not be completely correct as far as the technical rigour of the quantum field theory for Cosmology is concerned. So for our discussion we stick to the general models of inflations which are developed from more reliable version of UV complete theories at very high energy scale and these models has to have the reheating phenomenon in the evolutionary cosmological time scale of our universe). As a consequence of such interactions enormous amount of heat is generated and the quantum mechanical system that we are studying immediately goes to its out-of-equilibrium phase. From the previous understanding of the subject it was extremely difficult to determine the quantum effects in terms of studying the quantum correlations in this out-of-equilibrium phase. However, it is expected from our basic understanding of quantum statistical mechanics that if we wait for long cosmological time scale then the system reaches the equilibrium where it is possible to associate an equilibrium reheating temperature associated with the system. After reaching equilibrium the reheaton field started decaying to some new particle contents which are responsible to describe the genesis of dark matter in the early universe Cosmology. That means without explaining the detailed quantum aspects throughout the total cosmological time scale in detail it is impossible to study the genesis of the dark matter contents of our universe and this actually motivates us to study this hidden unexplored phenomenon in this paper. Until now we have very less description available regarding the quantum mechanical aspects during the epoch of reheating. Only the quadratic and quartic inflationary models are studied phenomenologically in this context and it is until unknown how to quantify and estimate the quantum mechanical correlations and study the detailed quantum field theory aspects of reheating. On the other hand, as we have already mentioned, it is extremely important to determine the quantum mechanical correlations during this epoch to study the genesis of the dark matter contents where the principles of out-of-equilibrium quantum field theory will play significant role. However, until now there are no such significant inputs available from the theory side which can be able to provide us the relevant tools and techniques to compute such quantum mechanical correlation function and study various other quantum field theory aspects related to reheating at out-of-equilibrium. In this paper, we have established a detailed quantum field theory framework at the out-of-equilibrium phase which helps us to quantify the extremely relevant quantum mechanical correlation functions in terms of OTOCs. The methodology presented in this paper not only helps us to quantify the correlation at the out-of-equilibrium phase, but also provides us a detailed understanding about the equilibrium behaviour of the correlation functions at a large time limit in the cosmological time scale. Additionally, this detailed study through the cosmological generalization of OTOCs helps us to determine a lower bound on the reheating temperature in a completely model independent way. From the previously available understanding of the subject it was possible to give an estimate of the reheating temperature in a completely model-dependent way after knowing about the relativistic degrees of freedom which are participating during the epoch of reheating. However, apart from having a model-dependent expression, that expression actually relies on the scale of inflation completely, which is not known from the observation side until now; only an upper bound on the scale of inflation in terms of the tensor-to-scalar ratio is known from the observational probes. So the previously described method of determining the reheating temperature is not very reliable and only helps us to know about the upper bound on the reheating temperature in a completely model-dependent way. On the other hand, the present study helps us to determine the exact time-dependent behaviour of the reheating temperature and additionally provides a lower bound of reheating temperature which we have derived in a completely model independent way. As we proceed through the subject material of the paper, one can have a clear understanding about each of our big claim which we have explicitly established with detailed analysis. Apart from this, one can consider another well known framework, which is described by the warm inflationary paradigm [145,146], where the radiation appears due to the stochastic particle production concurrently with inflationary expansion in the early universe evolutionary time scale. Sometimes this framework can be treated as the finite temperature generalization of usual well known zero temperature inflationary paradigm, which is mostly studied in cosmology literature. The presence of the effect of radiation during the epoch of inflation implies that there is a possibility exists on which the epoch of inflation could smoothly end directly into a radiation-dominated epoch, particularly without having a separate reheating epoch. This is one of the early universe phenomena apart from reheating where the effect of out-of-equilibrium physics is significantly dominating and the detailed quantum field theoretic phenomena have not been studied yet in detail. Until now, from the previous work, it is very clear that this phenomenon is very model-dependent and only a few models have studied this possibility. This is because of the fact that handling the out-of-equilibrium dynamics in the quantum regime is extremely complicated. However, we are again very hopeful that, similar to reheating, OTOC at finite temperature studied in this paper is able to describe the quantum correlation functions for the warm inflationary scenario, which is able to describe the out-of-equilibrium dynamical phenomenon in the quantum regime in a very elegant and model-independent way. Though it is true that our computation performed in this paper will be applicable to the warm inflationary scenario, we will restricted our analysis here to describe only the quantum effects during the epoch of reheating particularly in the very early time scale of evolutionary scale of our universe where the contribution of the out-of-equilibrium physics play pivotal role to control the dynamics.
- Stochastic inflation:Stochastic inflationary [147,148,149,150] paradigm is a very important aspect of the early universe Cosmology whose technical construction is completely different from the usual inflationary paradigm. To have inflation from a model which is derived from some UV complete high energy quantum field theories we do not need any additional source, time-dependent scalar inflaton field slowly rolls down through an inflationary potential and participate in the quantum field theory using which one can explicitly compute the N-point correlation functions within the framework of early universe Cosmology. However, in the framework of stochastic inflationary paradigm a time-dependent stochastic random source function play significant role to study the background construction of the quantum field theory. More precisely, here we have two fields, the inflaton and a stochastic random time-dependent noise field, where both of them are participating to carry forward a correct quantum field theory construction of inflationary paradigm within the framework of early universe Cosmology. To construct a correct and consistent version of these type of quantum field theories one needs to start with a theory where the time-dependent inflaton and the stochastic random time-dependent field coupled to each other. In most of the previous literature, during the construction of these type of quantum field theories the correlation between the time-dependent noise using which one can compute the N-point correlation function using the time-dependent inflation to study the role of quantum mechanical fluctuations in the early universe Cosmology. In this theoretical construction, the correlation between the noise acts as a source of the correlation between the inflaton. However, cannot say concretely about the exact behaviour of the noise at the starting point of the computation. Most of the computation until now have performed in this literature by assuming the underlying Gaussian behaviour of the probability distribution of the time-dependent stochastic noise function. As a consequence of this assumption one can now expect that the one-point function and any odd point correlation function of the time-dependent stochastic noise vanish trivially. On the other hand the two-point function has to de proportional to a time translationally invariant Dirac Delta function and the proportionality constant actually determine the amplitude of two-point correlation function, which we identify as the power spectrum of the time-dependent stochastic noise in the context of early universe Cosmology. One can also compute any other higher point even correlation functions from this construction, where one can explicitly show that the connected part of these correlations are factorizable in terms of the two-point function or the Green’s function of the theory which we are considering to describe the background quantum field theory of early universe Cosmology. If we believe that the initial assumption regarding having a Gaussian probability distribution of time-dependent stochastic noise is perfectly correct then everything we have mentioned above are the automatic consequence of that and using these information one can determine the N-point correlation function from the inflaton field consistently. Here the type of the noise we have pointed which follow the Gaussian probability distribution is commonly known as the white noise within the framework of quantum field theory. However, unfortunately we really do not have any idea if the assumption that we have taken at the starting point is correct at all or not when one can think of any arbitrary stochastic randomness within the framework of quantum field theory. This allows to think about considering non-Gaussian noise, which commonly identified as the coloured noise in the present context. However, if we have some time-dependent stochastic coloured non-Gaussian noise then it is extremely complicated to determine the quantum correlations between the noise and hence the N-point quantum correlation for inflaton field which is sourced by the coloured noise time-dependent profile. Furthermore, it is expected that the random quantum fluctuations in the stochastic noise is the prime source for which the background equilibrium quantum filed theory set up goes to its out-of equilibrium phase, where we have very less information regarding the cosmological correlation functions within the framework of quantum field theory of early universe Cosmology. This actually motivates us to think about some alternative construction of computing the N-point quantum correlation functions due to the presence of coloured time-dependent noise profile and in this paper by constructing the cosmological version of OTOC we have tried to address this crucial issue in an alternative way within the framework of out-of-equilibrium quantum field theory.
- Quantum quench in Cosmology:The study of quantum mechanical quench in the presence of time-dependent random coupling and its consequences in quantum correlation functions in Cosmology is a very important topic of study in the context of theoretical physics. Until now, this has not been very well understood and studied in the context of Cosmology. In the earlier literature for various statistical mechanical and condensed matter systems quantum quench has been studied rigorously, but its extension in the framework of early universe Cosmology will provide us the understanding of the thermalization phenomenon and the details of the achieving thermal equilibrium from an out-of-equilibrium phase in the presence of a random time-dependent coupling parameter. One can start with various possibilities here where in each case, theories are minimally coupled with classical FLRW conformally flat cosmological metric in a minimal fashion. The first and the simplest possibility appears in free scalar field theory where we consider a time-dependent mass, which is a random coupling parameter within the framework of quench. The second possibility appears within the framework of an interacting quantum field theory where one can consider a situation where two scalar fields with constant mass are interacting with each other in presence of a random time-dependent coupling parameter. If we treat the interaction term between the two scalar fields as quadratic, then constructing the effective theories of each scalar field becomes simpler after performing the path integration over the other unwanted scalar field for the description. In the quantum description, sometimes this is identified to be the partial trace operation when we are describing everything in terms of density matrix and similar approach have been followed in the context of quantum field theories driven by an open quantum system, where the system is non-adiabatically interacting with the environment. One can further generalize this idea for N number of scalar fields which are placed at the thermal bath and interacting with a system which is described by a singe scalar field. In terms of the interaction here one can consider the quadratic or any other non linear interactions. This description can be identified as the quantum field theory generalization of the well known Feynman–Vernon model of influence functional theories or the Caldeira Leggett model which describes the quantum dissipation phenomenon in the Quantum Brownian motion picture within the framework of early universe Cosmology. Considering these mentioned frameworks, one can explicitly compute the OTOCs using the methodology presented in this paper to study the quantum mechanical N-point out-of-time ordered correlation functions in presence of a quantum mechanical quench within the framework of out-of-equilibrium version [151,152,153,154] of open quantum field theory of Cosmology [155,156,157,158].
- Highlight I:The methodology presented in this paper helps us to quantify the quantum mechanical correlation functions within the framework of Cosmology in the presence of random quantum fluctuations. In this connection, we have computed the expressions for the two-point and four-point cosmological OTOC in the quantum regime which will provide the behaviour of the correlation functions in the out-of-equilibrium regime of the quantum field theory of early universe Cosmology.
- Highlight II:We have additionally studied the classical limiting version of the two-point and four-point cosmological OTOCs which will provide the decaying large time behaviour of the correlation function, which perfectly matches with the expectations from the chaotic phenomenon in the classical regime of the field theory.
- Highlight III:The large time limiting behaviour of the four-point OTOC helps us to comment on the equilibrium behaviour of the quantum correlations and to determine the lower bound of the equilibrium temperature. Using this concept, one can further determine the lower bound on the reheating temperature within the framework of early universe Cosmology.
- Highlight IV:We have also studied the quantum Lyapunov spectrum for Cosmology and computed the associated quantum Lyapunov exponent to have a consistent chaotic description in the quantum regime from the four-point cosmological OTOC derived in this paper.
- Highlight V:We have explicitly proved from our detailed computation that the results obtained from the normalized version of the four-point cosmological OTOC is completely independent of the choice of the time-dependent perturbation variable as appearing in the specific scheme of the cosmological perturbation theory.
- In Section 2, we discuss how one can formulate the OTOC in the context of Cosmology and what exact quantity we have to compute for this study.
- In Section 3, we discuss about the detailed computation of quantum micro-canonical two-point and four-point OTO amplitudes and the related OTOC in Cosmology.
- In Section 8, we discuss about classical limit of micro-canonical two-point and four-point OTO amplitudes and the related OTOC in Cosmology.
2. Formulation of OTOC in Cosmology
2.1. General Remarks on OTOC
2.2. Eigenstate Representation of OTOC in Quantum Statistical Mechanics
2.3. Constructing OTOC in Cosmology
- We are dealing with commutative version of the quantum field theory, i.e., all the field momenta and the fields are commutative amongst themselves. Things will change when one deals with the non commutative version of the quantum field theory which means in that in that case all the field momenta and the fields are non-commutative amongst themselves. Usually in both of the versions of quantum field theories we mostly consider the equal time commutation relations at different space points (ETCR) and the unequal time commutation relations (UETCR). In the Fourier transformed versions which can be translated as computing commutation relations at different momenta with same time for ETCR and with different time for UETCR. However, if we look into the specific mathematical structure of OTOC, then we see that the commutators are either ETCR or UETCR with fixed momentum or position.
- If we get divergence to compute OTOC, then like the previous cases we deform the contour of integration in Schwinger–Keldyshpath integral formalism by introducing one or more regulators in the theory. In the technical ground, introducing the regulator cut-off means one needs to deform the Schwinger–Keldyshpath integral contour at finite temperature by introducing a single or more than one parameter. Consequently, in that context we will get a finite regulated result. On the other hand, in the context of non-commutative version of the quantum field theory it is expected that the form OTOC (the mathematical definition of OTOC is provided in the next subsections explicitly for massless, partially massless and massive scalar field theories) we get non vanishing physically relevant answers. During the computation of the OTOC if we get some divergences then like previous case here we also deform the contour of integration by introducing a cut-off to get finite results.
- The explicit time-dependence of the OTOC is very important in the present context, particularly the study the physics of out-of-equilibrium. If the second term of the OTOC grows exponentially with time then one can quantify quantum chaos from the present computation and check whether the final result saturated the chaos bound on the Lyapunov exponent or not. On the other hand, if instead of getting an exponentially growing time-dependent behaviour if we get some other non-trivial characteristic behaviour as a function of time that is also very important to study in the present context to study the out-of-equilibrium behaviour of quantum mechanical systems considered in the framework of Cosmology. Particularly in Cosmology, studying the non trivial time-dependent behaviour of the OTOC for stochastic particle production during inflation, reheating are very useful to extract physical information from the quantum mechanical systems studied. Since we are trying to compute the OTOC for quantum field theory, it is expected to get some interesting physical outcomes from the correlation functions. However, it is also expected from the studies of the OTOC that at very large time limit the correlation functions reach thermal equilibrium and saturates to a finite value. This may not indicate the saturation of the quantum chaos, but this will also play significant role to study the time-dependent behaviour of OTOC. In this context, one can use this information as a limiting boundary condition of OTOC. Apart from that the early time behaviour of OTOC is also good to know. The early time behaviour is physically important as it gives the idea about the initial condition on the time-dependent behaviour of the OTOC. Initial condition OTOC actually fixes from which point in the time scale the quantum system goes to out-of-equilibrium. This information in the context of Cosmology is very useful because it will give a physical picture about a particular out-of-equilibrium phenomenon in which we are interested.
- In this computation, we are considering massless (), partially massless () and massive () scalar field for the computation of OTOC in the context of Cosmology. In the context of Cosmology, the fundamental characteristic scale is described by Hubble scale, i.e., H. Now compared to this characteristic scale we define massless, partially massless and massive scalar field in the de Sitter gravitational background. In the context of massive scalar field theory one usually consider various time-dependent protocols to describe the stochastic particle production phenomenon during inflation and reheating process in the context of primordial Cosmology. For a few types of time-dependent protocols one can exactly analytically solve the field equations for the scalar field, which are commonly used in the study of quantum quench. In general exact solution of the field equation is very difficult so solve and for this purpose one use the WKB approximation method to find out the analytical solution of the scalar field. These solutions are extremely important to study in de Sitter cosmological background as it provide the key ingredient to compute the previously mentioned OTOC in the present context. In the context of stochastic particle production one consider Dirac Delta type of scatterer as a time-dependent protocol, which is nothing but the white noise in the present context. In some cases, one also considers coloured noise time-dependent protocol. In the quantum description, the white noise is identified to be non-Markovian and coloured noise is considered as Markovian at the level of correlation function (quantum analogue: at the level of quantum correlation function for the white and coloured noise are described by the following correlation functions:
2.3.1. For Massless Scalar Field
2.3.2. For Partially Massless Scalar Field
2.3.3. For Massive Scalar Field
3. Quantum Micro-Canonical OTO Amplitudes and OTOC in Cosmology
3.1. Computational Strategy
- At first, we need find out the analytic solution of the equation of motion of the scalar field in the corresponding FLRW flat spatial background.
- From the above mentioned solution we need to find out the canonically conjugated field momentum and using that we need to compute the square of the quantum mechanical commutator bracket. It is proved during performing this operation we need to quantize the field content in terms of the creation and annihilation operators.
- From the above mentioned result we need to further compute the thermal average value of the square of the commutator bracket of the field and its canonically conjugate momenta, which physically represents the four-point out-of-time ordered correlation function.
- Next, we need to compute the partition function the quantum system under consideration by computing the thermal Boltzmann factor and the trace of the corresponding thermal Boltzmann factor. To serve this purpose we need to also compute the expression for the Hamiltonian from the quantum mechanical system under consideration.
- Furthermore, we need to compute the thermal two point out-of-time ordered correlation functions from the field content and the corresponding canonically conjugate momentum. This will help us to find out the expression for the normalised OTOC which is defined in the earlier sections explicitly.
- Furthermore, instead of computing these description in the coordinate space one can do a similar computation of OTOC in Fourier space as well. For this purpose one needs to compute all of the required operators in the Fourier transformed space. Sometimes doing the computation in Fourier transformed space is simpler than the computation in coordinate space. Most importantly, doing the computation in Fourier space is really very helpful in the context of Cosmology. The primary reason is that doing the computation in momentum space (which is the Fourier transformed version of the coordinate space) in the context of Cosmology is more technically interesting. Actually, in momentum space most of the quantum correlation functions in the context of Cosmology preserves the conformal symmetry under conformal transformations. In quasi de Sitter space, this conformal symmetry is slightly broken in Fourier space by the amount of slow-roll parameters, which we have taken into account in our computations.
- In the present context it is better to perform the whole computation in a preferred choice of gauge, which is commonly performed in the context of Cosmological Perturbation Theory. Following the same strategy here we work on gauge, which makes the computation of OTOC simpler than doing the computation doing without the choice of a preferred gauge.
- It is important to mention here that doing the computation of OTOC for N interacting scalar field analytically are extremely complicated. On the other hand, doing the computation with N non-interacting multi scalar field is also not very simple. For this reason we try in this paper to restrict ourself to the non-interacting single field and N field case. For the single field case we also provide the computation of OTOC in the previously mentioned preferred gauge choice.
- Last but not least, doing the computation of OTOC in any arbitrary dimensions is not at all possible analytically. So we have to restrict our computation in this article in spatial dimension which is compatible with our spatially flat dimensional space time in FLRW cosmological background.
3.2. Classical Mode Functions in Cosmology
3.3. Quantum Mode Function in Cosmology
- In the de Sitter case with FLRW background is a slowly varying conformal time-dependent function in general, which physically represents a very slight deviation from the de Sitter solution, where the Hubble parameter is dominated by Cosmological Constant in Planckian units. It is observed from the computation and that both of the parameters capture the effect of such deviation and for this reason sometimes one refers this as the quasi de Sitter space-time.
- Another important fact is that since we strictly do not know the explicit conformal tie dependence of the reheating equation of state , we have assumed that the time-dependence is very slow and for this purpose, this crucial parameter exactly behaves like a constant with the restriction that . This fact is extremely important to mention as it fixes the form of the scale factor , parameter (the slow-roll parameter is commonly used in the context of quasi de Sitter inflationary phase where the scale factor can be expressed as . During inflation and at the end of inflationary phase , which is very useful to determine the corresponding field value at the end of inflation. So during stochastic particle production during the inflationary phase, we will respect these constraints on the slow-roll parameter . On the other hand, since reheating happened after inflation then it is expected that the magnitude of the parameter during reheating, where the scale factor can be expressed as . This statement can be justified very easily in the context during the epoch of reheating. From the previously derived expression for the parameter in the context of reheating we have found that , where the equation of state parameter . This directly implies that during the epoch of reheating the parameter will lie within the window, (alternatively one can say that the Hubble velocity is lying within the window in the Planckian units), which is obviously a very strong and specific information we get from the from the present computation) and the Mukhanov–Sasaki variable in terms of conformal time as well as the equation of state parameter in the present context.
3.4. Canonical Quantization of Cosmological Hamiltonian: Classical to Quantum Map
3.5. Cosmological Two-Point and Four-Point “In-In" OTO Micro-Canonical Amplitudes
3.5.1. OTOC Meets Cosmology
- Information I:First of all, one need the quantum operators corresponding to the rescaled field and its canonically conjugate momenta, which is actually written in terms of the classical solution obtained from the mode equation and the creation and annihilation operators. Actually, in the context of Cosmology one can write down the Hamiltonian of the system in terms of an Harmonic oscillator with conformal time-dependent frequency in a Fourier space and after constructing this Hamiltonian one needs to integrate over all possible momenta. Using this prescription and using the canonical quantization procedure one can easily quantize the system Hamiltonian in the framework of Cosmology.
- Information II:Next, one needs to compute the expression for the expression for the square of the commutator bracket in the background of FLRW space-time in curved space quantum field theory in coordinate space. This is the prime component which will fix the final expression for OTOC in the framework of Cosmology. We will explicitly show that this contribution can be written in terms of four parts where each of the parts in Fourier transformed space mimics the role of some kind of scattering amplitudes after taking its trace, which are functions of four momenta and two time scales as appearing in the definition of OTOC. Precisely these four momenta are coming due to the fact that in each contribution, we are dealing with four quantum operators. Now if we are careful about the technicality then we have to say these four-point scattering amplitudes are basically the four-point time-dependent correlation functions in the framework of Cosmology as during performing the trace operation we have to use the same quantum vacuum state, which means that the initial and final state both are the same, and identified as the “in” state. So it is basically an in-in amplitude rather than an usual “in-out” amplitude. The usual “in-out” amplitude can be usually computed using the idea of S matrix, which is a Schwinger Dyson series in the context of quantum field theory and this is an unitary matrix in the context of a closed system in our Universe when the system under consideration is not exchanging any information with the surroundings. On the other hand, in the Cosmological set-up we deal with “in-in” amplitudes which can be computed using the well known Schwinger–Keldyshformalism. Instead of calling this quantity which we want to evaluate as “in-in” amplitude we call these contributions “in-in” quantum correlation functions in the framework of Cosmology.
- Information III:Furthermore, we have to fix the definition of trace in the context of quantum field theory in a classical gravitational background, specifically in the context of FLRW Cosmology. To serve this purpose, we need to first use a standard definition of the quantum wave function of our Universe. For this purpose, the most common choice is to use the definition of standard Bunch–Davies vacuum state, which is basically a thermal ground state in the framework of Cosmology. Sometimes this vacuum is identified to be the Hartle–Hawking or Cherenkov vacuum state in quantum field theory theory of curved space-time with Cosmological background. Apart from that, another useful vacuum is the vacuum, which is mostly used in the context of interacting quantum field theory in curved space-time with cosmological background. Here in the construction of the vacuum it appears as a one real parameter family which can take any continuous value starting from the zero value. The zero value is a special case of the vacuum which is basically the well known Bunch–Davies quantum vacuum ground state. One the other hand, for the other values of one can construct infinite number of states which can be considered as some excited states which can be expressed in terms of the known Bunch–Davies vacuum state using the Bogoliubov transformation.
- Information IV:Using the above mentioned vacuum or the Bunch–Davies quantum vacuum one can further compute the numerator of OTOC, which is the trace of the square of the commutator bracket of the rescaled field variable and its canonically conjugate momenta along with the thermal Botzmann factor in which the system Hamiltonian is appearing explicitly. Furthermore, one can compute the denominator of the OTOC which represents the expression for the trace of the only thermal Boltzmann factor, which is physically identified to be the thermal partition function in the Cosmological framework. Then putting a negative sign in front of the computed object, one can derive the expression for the OTOC in un normalized form. This overall negative sign is very important as it makes the un normalized OTOC positive and growing with the combined time scale in which both of the above mentioned quantum extended operators are defined. In this computation, the thermal partition function and the trace of the commutator bracket square at finite temperature are evaluated using semi classical approximation, which means that we treat the fluctuation of scalar modes appearing from cosmological perturbation theory in the primordial universe. Precisely the perturbation in the metric in the FLRW background can be written in terms of scalar, vector and tensor modes in Fourier transformed space. This is commonly known as the SVT decomposition in Cosmology. In this computation, we have restricted our attention to scalar modes which can promoted to be a quantum operator during the computation of OTOC. It is obvious that the canonically conjugate momenta computed from the scalar modes can also be promoted to be a quantum mechanical operator. However, if we look into this problem very carefully, then we observe that the origin of the quantum fluctuations from the scalar modes are coming from metric perturbation in the FLRW background, which we actually treat purely classically. For this reason we will do a semi-classical (not purely quantum or classical) computation for the computation of OTOC in the framework of primordial cosmology.
- Information V:Last but not least, we have to fix the normalization of OTOC. This can be perfectly done using the previously mentioned thermal trace operation in presence of vacua or Bunch–Davies quantum vacuum state in the context of Cosmology. After normalization we can able to construct a dimensionless cosmological four-point “in-in" amplitude or quantum OTOC. We have computed the normalization factor in the Appendix A, Appendix B, Appendix C, Appendix D, Appendix E, Appendix F, Appendix G, Appendix H, Appendix I, Appendix J, Appendix K, Appendix L and Appendix M in detail. Please look into the technical details in the Appendix C, Appendix D and Appendix M.2 respectively.
3.5.2. Fourier Space Representation of the Commutator Bracket: Application to Two-Point OTOC
3.5.3. Fourier Space Representation of Square of the Commutator Bracket Application to Four-Point OTOC
3.6. Cosmological Thermal Partition Function: Quantum Version
3.6.1. Quantum Vacuum State in Cosmology
3.6.2. Quantum Partition Function in Terms of Rescaled Field Variable
3.6.3. Quantum Partition Function in Terms of Curvature Perturbation Field Variable
3.7. Trace of Two-Point “In-In" OTO Amplitude for Cosmology
3.8. OTOC from Regularised Two-Point “In-In" OTO Micro-Canonical Amplitude: Rescaled Field Version
3.9. OTOC from Regularised Four-Point “In-In" OTO Micro-Canonical Amplitude: Curvature Perturbation Field Version
- First, we have explicitly shown that the definition of the two-point OTOC is completely coordinate-independent and just only depend on the time scale on which the operators in cosmological perturbation theory are time scale separated. Even if we start defining the operators in a specific space point at different times, in the final result all such information is integrated out and we get completely a time-dependent two-point OTOC.
- Next, it is important to note that for , which implies , the two-point OTOC trivially vanishes. However, for partially massless and heavy scalar fields, two-point OTOC becomes non-trivial and carries significant information of the theory.
- We also have found that the final obtained result of the two-point OTOC is -independent even though we start with a thermal micro-canonical statistical ensemble during definition the two-point OTOC in the trace formula.
- Furthermore, we observe that if we fix , that means if both the quantum operators representing the cosmological fluctuations are defined at the same time then two-point OTOC explicitly gives diverging contribution in the quantum regime. This is perfectly consistent with the definition of any general OTOC and also supports all the physical requirements to construct the foundational set-up to compute OTOC.
- If we perform an explicit computation, then it is possible to show that in our set-up the one-point functions in the context of Cosmology are given by the following expressions:The similar statement is also valid if we define the one-point functions for Cosmology in terms of the curvature perturbation field variable and its canonically conjugate momentum. This result implies that there is no point of performing normalization of the obtained two-point OTOC for Cosmology with respect to the above mentioned one-point functions. So technically, it is sufficient enough to compute the two-point OTOC without normalization in the context of Cosmology.
- Another important point we have to mention is that not only the one point function but also the three point functions in the present context is trivially zero which can be explicitly proved by making use of the well known Kubo–Martin–Schwinger condition in terms of time translational symmetry of the thermal correlation functions computed for Cosmology. These possibilities are explicitly mentioned below: