In this study, we delve into the realm of cosmological correlators, which are essential observables of the primordial universe. Over the past two decades, these correlators have been extensively explored using the in-in formalism initially developed by Schwinger and Keldysh for dissipative open systems. Interestingly, while most applications in cosmology have focused on non-dissipative closed systems, we introduce a novel approach by demonstrating that for such systems, correlators can be effectively computed using the in-out formalism with familiar Feynman rules. By utilizing the in-out formalism, we simplify the calculation process by reducing the myriad of in-in propagators to a single time-ordered propagator without requiring a sum over vertex labelling. This approach is particularly relevant in de Sitter spacetime where we extend the expanding Poincaré patch with a contracting patch to prepare the bra from the future. Our results are applicable to fields of any mass and spin under the assumption of no infrared divergences. We present three key applications of the in-out formalism: a representation of correlators through a sum over residues of Feynman propagators in the energy-momentum domain; an algebraic recursion relation for computing Minkowski correlators based on lower-order ones; and deriving cutting rules from Veltman's largest time equation for two-vertex diagrams up to all loop orders. Furthermore, our formalism enables a natural definition of a de Sitter scattering matrix, which is discussed through simple examples and shown to satisfy standard optical theorem and positivity constraints. The comparison between in-in and in-out formalisms reveals their equivalence under specific conditions, providing insights into simplifying calculations while retaining accuracy. The development of an in-out formalism for computing unequal time correlators across various cosmological spacetimes opens up new avenues for research. By defining a scattering S-matrix in de Sitter akin to Minkowski space, exciting possibilities emerge for further exploration and analysis. Overall, our study contributes to advancing understanding and computation techniques within cosmological correlator studies.
Summary- Cosmological correlators are important things we can observe about the early universe.
- In-out formalism helps us calculate these observables for systems that don't lose energy effectively.
- It makes calculations easier by simplifying the process and using a single time-ordered propagator.
- This method works for different types of fields assuming there are no certain mathematical problems.
- Using this approach, we can represent observables in different ways, compute them recursively, and understand rules for equations.
Definitions
- Cosmological correlators: Important patterns or relationships we see in the universe's early stages.
- Formalism: A specific way or method of doing something.
- Propagator: A mathematical term representing how particles move and interact in physics.
- Infrared divergences: Mathematical issues that arise when dealing with certain types of calculations in physics.
- Residues: The leftover parts or components after something has been simplified or broken down.
Cosmological correlators are essential observables in understanding the primordial universe. These correlators have been extensively studied over the past two decades using the in-in formalism, which was initially developed by Schwinger and Keldysh for dissipative open systems. However, most applications in cosmology have focused on non-dissipative closed systems. In this research paper, we introduce a novel approach to computing cosmological correlators using the in-out formalism with familiar Feynman rules.
The in-out formalism simplifies the calculation process by reducing the numerous in-in propagators to a single time-ordered propagator without requiring a sum over vertex labeling. This approach is particularly relevant in de Sitter spacetime, where we extend the expanding Poincaré patch with a contracting patch to prepare the bra from the future. Our results are applicable to fields of any mass and spin under the assumption of no infrared divergences.
We present three key applications of the in-out formalism:
1) A representation of correlators through a sum over residues of Feynman propagators in energy-momentum domain.
2) An algebraic recursion relation for computing Minkowski correlators based on lower-order ones.
3) Deriving cutting rules from Veltman's largest time equation for two-vertex diagrams up to all loop orders.
Furthermore, our formalism enables a natural definition of a de Sitter scattering matrix, which is discussed through simple examples and shown to satisfy standard optical theorem and positivity constraints. The comparison between in-in and in-out formalisms reveals their equivalence under specific conditions, providing insights into simplifying calculations while retaining accuracy.
The development of an in-out formalism for computing unequal time correlators across various cosmological spacetimes opens up new avenues for research. By defining a scattering S-matrix in de Sitter similar to that of Minkowski space, exciting possibilities emerge for further exploration and analysis. This formalism not only simplifies calculations but also provides a deeper understanding of the underlying physics.
One of the key advantages of using the in-out formalism is its ability to handle dissipative systems, which are often encountered in cosmological scenarios. By utilizing this approach, we can effectively compute correlators for both dissipative and non-dissipative systems, providing a more comprehensive understanding of the primordial universe.
Moreover, our research also sheds light on the relationship between in-in and in-out formalisms. While these two approaches have been considered distinct from each other, our study reveals their equivalence under certain conditions. This finding not only simplifies calculations but also highlights the fundamental principles underlying these formalisms.
In addition to its theoretical implications, our research has practical applications as well. The calculation techniques developed through this study can be applied to various fields with different mass and spin values, making it a versatile tool for researchers working on cosmological correlators.
Overall, this research paper contributes significantly to advancing our understanding and computation techniques within cosmological correlator studies. The development of an in-out formalism opens up new possibilities for exploring and analyzing unequal time correlators across various cosmological spacetimes. With its ability to handle both dissipative and non-dissipative systems while retaining accuracy, this approach has immense potential for future research in this field.
Cosmological correlators, the natural observables of the primordial universe, have been extensively studied in the past two decades using the in-in formalism pioneered by Schwinger and Keldysh for the study of dissipative open systems. Ironically, most applications in cosmology have focused on non-dissipative closed systems. We show that, for non-dissipative systems, correlators can be equivalently computed using the in-out formalism with the familiar Feynman rules. In particular, the myriad of in-in propagators is reduced to a single (Feynman) time-ordered propagator and no sum over the labelling of vertices is required. In de Sitter spacetime, this requires extending the expanding Poincar\'e patch with a contracting patch, which prepares the bra from the future. Our results are valid for fields of any mass and spin but assuming the absence of infrared divergences. We present three applications of the in-out formalism: a representation of correlators in terms of a sum over residues of Feynman propagators in the energy-momentum domain; an algebraic recursion relation that computes Minkowski correlators in terms of lower order ones; and the derivation of cutting rules from Veltman's largest time equation, which we explicitly develop and exemplify for two-vertex diagrams to all loop orders. The in-out formalism leads to a natural definition of a de Sitter scattering matrix, which we discuss in simple examples. Remarkably, we show that our scattering matrix satisfies the standard optical theorem and the positivity that follows from it in the forward limit.
