Excitability of Gaussian states with VEVs

Technical Summary: Excitability of Gaussian States with VEVs

Problem Statement
In algebraic quantum field theory (AQFT), the concept of "excitability" addresses whether one state $\omega_2$ can be locally excited out of another reference state $\omega_1$. This is determined by whether $\omega_2$ admits a representative density matrix $\rho_{\omega_2}$ within the GNS Hilbert space $\mathcal{H}_{\omega_1}$ of the reference state. While necessary and sufficient criteria for excitability were previously established for zero-mean Gaussian states in generalized free field theories [1], a gap remained for Gaussian states with non-vanishing one-point functions (Vacuum Expectation Values or VEVs). Such states, including generic coherent states and specific holographic states, are physically relevant but require a generalized framework to determine if their excitations are accessible from a given vacuum.

Methodology and Toolkit
The authors extend the algebraic framework of [1] to accommodate non-zero mean states. The methodology relies on the following key technical components:

1. Gaussian State Formalism: The analysis utilizes the property that Gaussian states are fully characterized by their one-point functions (VEVs) and connected two-point functions. The connected two-point functions define a positive semi-definite inner product $\mu$ on the space of smearing functions, which determines the topology of the GNS Hilbert space.
2. VEV Shifting: The authors introduce a formal operation to shift the one-point function of a Gaussian state $\omega$ by a linear functional $v$, denoted $\omega^v$, while preserving the connected two-point functions.
3. Canonical Purification: To handle mixed states, the paper employs the technique of canonical purification [9]. Any state $\omega$ can be embedded into a "centrally pure" state $\hat{\omega}$ (a purification on a doubled algebra). Excitability is preserved under this purification, allowing the authors to reduce general mixed-state problems to the study of centrally pure states.
4. Decomposition of the Problem: The proof strategy decomposes the general excitability condition into two distinct lemmas:
   • Lemma 1: Excitability is invariant under simultaneous VEV shifts applied to both the reference and target states. This is proven via the construction of an intertwining unitary map.
   • Lemma 2: The condition for exciting a VEV shift $v$ out of a zero-mean Gaussian state $\omega$. This is proven by analyzing pure states, abelian algebras, and centrally pure states, utilizing the Riesz representation theorem and properties of the commutator operator $R$.

Key Contributions and Results
The paper establishes Theorem 1, which provides necessary and sufficient conditions for a Gaussian state $\omega_2$ to be excitable from $\omega_1$ ($\omega_2 \prec \omega_1$):

1. Zero-Mean Condition: The zero-mean counterparts of $\omega_1$ and $\omega_2$ (defined by their connected two-point functions) must be mutually excitable. This recovers the criteria established in [1].
2. VEV Boundedness Condition: The difference between the one-point functions, $v_{12}[f] = \omega_2(\phi[f]) - \omega_1(\phi[f])$, must be $\mu_1$-bounded. Specifically, there must exist a constant $C$ such that $|v_{12}[f]|^2 \leq C \langle f|f \rangle_{\mu_1}$ for all smearing functions $f$.

The authors prove that if the VEV difference is unbounded, no density matrix representative exists in the reference Hilbert space. Conversely, if the VEV difference is bounded, the excitability of the full states is equivalent to the excitability of their zero-mean parts.

Application to Anti-de Sitter (AdS) Spacetime
As a concrete application, the authors analyze free Klein-Gordon fields in the Poincaré patch of AdS space. They demonstrate that:

• A VEV shift $v_s$ can be excited from the AdS vacuum if and only if the mode profile of the shift, $\tilde{v}_s(k)$, is square-integrable ($L^2$) with respect to the momentum measure defined by the bulk inner product.
• This bulk condition is shown to be equivalent to the excitability of the corresponding boundary VEV in the dual Conformal Field Theory (CFT). The boundary VEV, defined via the extrapolate dictionary, is excitable from the CFT vacuum if and only if the bulk VEV is excitable from the AdS vacuum.

Significance and Claims
The paper claims to generalize the algebraic understanding of state excitability to the physically ubiquitous case of non-zero mean Gaussian states. The primary significance lies in:

• Rigorous Criteria: Providing a mathematically precise condition (VEV boundedness relative to the two-point function topology) that complements the existing criteria for zero-mean states.
• Holographic Consistency: Demonstrating that the excitability of bulk coherent states in AdS is strictly tied to the excitability of their boundary duals. This supports the consistency of the bulk-boundary duality in the context of state space structure, showing that failures of excitability in the bulk (due to infrared effects) are mirrored in the boundary theory.
• Generalization of Pure State Results: Extending known results for pure coherent states [10, 11] to arbitrary mixed Gaussian states, including those with non-trivial centers (centrally pure states).

The authors maintain a modest scope, focusing strictly on the algebraic criteria for excitability within generalized free field theories and their application to AdS/CFT, without proposing new experimental protocols or extending the results to interacting theories beyond the scope of the Gaussian approximation.