Abstract: Microcausality -- the vanishing of commutators outside the lightcone -- is a fundamental property of relativistic quantum field theories. We derive its implications for two-point functions of scalar operators on {\it Lorentz-breaking} states. We restrict to spatially homogeneous and isotropic states, at zero and finite temperature, such as finite-density states of matter and primordial inflationary states. In a mixed $(t, \vec k \, )$ representation, we find certain analyticity and exponential boundedness conditions, which we verify in a variety of examples. Crucially, we discuss how our conditions can be tested within the regime of validity of Lorentz-breaking low-energy effective field theories, clarifying the role of the group velocity of low-energy excitations. In the cosmological case, we derive a positivity condition on an EFT coefficient in an inflationary background. Lastly, we comment on how microcausality can be used to constrain higher-point correlation functions, via suitable nested commutators.
The paper shows that microcausality persists in Lorentz-violating contexts by deriving analyticity and exponential boundedness conditions for scalar two-point functions.
It employs a mixed (t,⃗k) representation to analyze correlators in various states, including finite temperatures and densities, thereby validating operator-level microcausality.
The findings impose constraints on effective field theories in cosmology and non-relativistic systems, ensuring consistency with subluminal group velocities and underlying quantum principles.
Exploring Microcausality under Lorentz Violation
The paper "Microcausality without Lorentz invariance" explores the foundational aspect of microcausality—a crucial property of relativistic quantum field theories, which stipulates that local observables commute at spacelike separations. This concept is typically intertwined with Lorentz invariance, yet it holds universally, even in Lorentz-breaking scenarios, as an operator statement. The researchers explore the implications of microcausality on two-point functions of scalar operators in states that lack Lorentz symmetry, with the aim of determining conditions imposed on such systems.
In particular, the study focuses on spatially homogeneous and isotropic states at both zero and finite temperatures, including finite-density states of matter and primordial inflationary states. The primary objective is to evaluate the consequences of microcausality within Lorentz-breaking frameworks from an effective field theory (EFT) perspective.
Methodological Insights
The authors scrutinize two-point correlation functions using a mixed representation approach, considering both the temporal and spatial (Fourier) domain simultaneously. By using this mixed (t,k) approach, they derive analyticity and exponential boundedness conditions for two-point functions that preserve microcausality.
Key findings include:
Analyticity: The two-point functions remain analytic in the complex spatial momentum k domain across any Lorentz-breaking conditions considered.
Exponential Boundedness: These functions are rigorously bound by an exponential factor, dependent on the geometry of the configuration in k space, ensuring the physical principle of microcausality across various setups.
Implications and Spectral Analysis
In the field of cosmology, the research yields constraints on effective field theories, particularly those related to inflationary models. A noteworthy implication in cosmological models is the derivation of positive bounds on an EFT coefficient in an inflationary background when examined under these conditions.
Moreover, their analysis has implications for assessing the applicability of Lorentz-breaking low-energy effective field theories. They contend that microcausality can indeed be tested in this regime, an assertion that counters some claims in the current literature that dismiss the possibility of microcausal validation within the constraints of low-energy theories.
Results from Specific Phenomenological Cases
The paper also explores specific contexts such as:
Relativistic Free Scalar Fields: Numerical checks show analyticity in complex k2 and compliance with exponential boundedness, even in interacting relativistic QFT applications.
Non-relativistic Effective Field Theories (NREFT): Microcausality compliance was confirmed with mass equivalent constraints on effective field theory cutoffs, maintaining subluminal group velocities.
Superfluid Phonons: Through scrutiny of UV completions and superfluids with broken Lorentz invariance, the terms dictating dispersion relations show the necessity of constraints like subluminal sound speeds.
Overall, this paper charts new territories in understanding the maintenance of fundamental quantum mechanical properties like microcausality in realms where Lorentz invariance is not presumed. It provides a rigorous framework within which effective field theories can be evaluated for consistency with deeper quantum field principles even in unconventional and complex scenarios. These insights significantly contribute to ongoing research in quantum field theory and cosmology, particularly in examining early universe conditions and state metamorphoses beyond standard models.