Unruh-DeWitt Type Coupling in GBF

• Unruh-DeWitt type coupling is a localized interaction between a quantized field and a two-level detector, used to probe the vacuum structure in quantum field theory.
• It operationalizes vacuum selection by enforcing the no-click condition, uniquely determining the complex structure for propagating modes and highlighting nonlocal effects for evanescent modes.
• Within the General Boundary Formulation, this framework connects detector responses with boundary data, providing practical insights for modeling quantum fields in regions with external influences.

An Unruh-DeWitt (UDW) type coupling is a localized interaction between a quantized field and an idealized two-level system (or harmonic oscillator), widely used to model the operational detection of field quanta, probe vacuum response, and analyze quantum information exchange in relativistic and boundary-aware field theories. The UDW coupling serves as an operational bridge between quantum field theory and “particle detection,” with a coupling form that ensures both technical robustness and flexibility for extracting properties of quantum fields in diverse spacetime settings. In the General Boundary Formulation (GBF), the coupling’s impact on vacuum selection and its dependence on external boundary conditions can be characterized precisely, providing insight into the nonlocal structure of the quantum field vacuum.

1. Principles and Mathematical Structure of the UDW Coupling

The prototypical UDW coupling is given by a bilinear interaction between a field operator and a detector operator along a prescribed worldline. In the GBF context, the detector is typically a harmonic oscillator with coordinate $q(\tau)$ following a trajectory $\gamma(\tau)$ and linearly coupled to a scalar field $\phi(x)$: $$\mathcal{V}_{UD}(\phi, q) = \lambda \int d\tau \, \phi(\gamma(\tau))\, q(\tau)$$ where $\lambda$ is a coupling constant. The system’s total action becomes

$$S[\phi, q] = S_0[\phi] + S_0[q] + \mathcal{V}_{UD}(\phi, q)$$

The first-order transition amplitude for the detector to become excited is proportional to a trajectory-averaged one-point function: $$\rho_{BD} = i \lambda \sqrt{\frac{2}{m\Omega}}\int d\tau e^{i\Omega\tau} \rho^{(\phi(\gamma(\tau)))}[|\psi\rangle \otimes |0\rangle]$$ where $m$ and $\Omega$ are the detector’s mass and frequency.

Transition probabilities for ground-to-excited ($g\!\to\!e$) and excited-to-ground ($e\!\to\!g$) transitions over an interval $[\tau_1,\tau_2]$ take the form

$$P^{(\Omega, \xi)}(g \to e) = \frac{2\lambda^2}{m\Omega} \int_{\tau_1}^{\tau_2} d\tau d\tau'\, e^{i\Omega(\tau-\tau')}\, \hat{\xi}(\gamma(\tau))\, \overline{\hat{\xi}(\gamma(\tau'))}$$

where $\hat{\xi}$ is a boundary field configuration “lifted” via the complex structure $J$ that determines positive frequency modes.

2. Vacuum Selection and the Role of the Complex Structure

Within the GBF, quantum states are expressed as holomorphic functionals over the boundary solution space, where the vacuum is not selected by an intrinsic time-evolution but by a complex structure $J$ imposed on the classical phase space at the relevant boundary. The UDW detector provides an operational criterion to fix this structure. The “no-click” condition requires that a detector at rest, coupled to a field prepared with only outgoing (energy-flux-defined) modes, must not become excited. Explicitly, for field configurations $\xi_{p_0}$, the positive-flux modes $L^{\mathbb{C}}_+$ are those for which

$$-n_\mu T^{\mu 0}(\operatorname{Re}\,\xi_{p_0})(x) > 0$$

where $n_\mu$ is the outward normal at the timelike boundary $\partial M$, and $T^{\mu\nu}$ the stress-energy tensor. The requirement is that the complex structure $J$ be chosen so that $P^{(\Omega, \xi)} = 0$ for any state $\xi$ composed solely of modes in $L_+^{\mathbb{C}}$. For standard oscillatory (propagating) modes in 1+1 Minkowski space, this fixes

$$J_{\Sigma_L} = -J_{\Sigma_R} = \frac{\partial_{x_1}}{\sqrt{-\partial_{x_1}^2}}$$

thus reproducing the usual split into positive- and negative-frequency components.

3. Evanescent Modes and the Need for Boundary Data

For evanescent modes ($p_0^2 - m^2 < 0$), which arise naturally in the presence of boundaries such as dielectrics, the spatial dependence becomes exponential. The general solution can be expressed as

$$\eta(x_0, x_1) = \int \frac{dp_0}{(2\pi)^3 2p_1} \left[\eta(p_0)f(p_0, x_1)e^{-ip_0 x_0} + \text{c.c.}\right]$$

with $$f(p_0, x_1) = \cosh(p_1(x_1 - \zeta)) + i \sinh(p_1(x_1 - \zeta))$$ for $p_0^2 - m^2 < 0$.

For these non-propagating modes, the “positive-flux” criterion depends on the position of the boundary, and the “no-click” condition does not uniquely determine $J$. Instead, to reproduce the physical detector’s response, the complex structure for evanescent modes must encode information about the specific matching conditions/physics outside the spacetime region $M$ (such as the location $\zeta$ of a dielectric interface and its refractive index $n$). The explicit form of the mode functions then becomes

$$e_{p_0; l}(x_1) = c_{a; l}(p_0)\cosh(p_1 x_1) + c_{b; l}(p_0)\sinh(p_1 x_1)$$

with coefficients such as $$c_{a; l}(p_0) = \cosh(p_1 \zeta) + i (p/(\sqrt{n} p_1))\sinh(p_1 \zeta)$$, revealing explicit dependence on external parameters (Banisch et al., 2012).

4. Operational Framework: Detector as Probe for Boundary State

In the GBF language, the correct selection of the field’s boundary state (i.e., the complex structure, and thus the vacuum) is obtained operationally through the UDW detector’s response. The process is:

• Formulate amplitudes for the field-detector system with respect to boundary data.
• Require that modes with strictly positive outgoing energy-flux (as determined by $T^{\mu\nu}$) result in a vanishing detector transition amplitude.
• For propagating modes, this uniquely fixes $J$ to the canonical positive-frequency prescription.
• For evanescent modes, compute the detector response with respect to the external physical context; only with knowledge of the outside (dielectric, boundary conditions, embedding geometry) can the correct $J$ be imposed.

This reveals the nonlocal character of vacuum selection in the presence of boundaries or inhomogeneities: the vacuum state defined on a finite spacetime region must depend on global boundary physics, consistent with established manifestations of vacuum nonlocality such as the Reeh-Schlieder theorem.

5. Summary Table: Core Features of UDW-Type Coupling in the GBF

Aspect	Propagating Modes (Oscillatory)	Evanescent Modes (Exponential)
Complex structure $J$ fixing	No-click/outgoing-flux $\Rightarrow$ unique	No-click condition insufficient; requires external data
Vacuum dependence	Locally determined (intrinsic to $M$)	Nonlocally determined (depends on exterior/boundaries)
Boundary state operational tool	UDW detector as “vacuum probe”	Requires embedding or boundary physics in model

For propagating modes, the UDW coupling provides a unique prescription for vacuum selection via the no-click condition and symmetry. For evanescent modes, the correct operationally meaningful vacuum (complex structure) is only definable by reference to global information.

6. Physical Implications for Quantum Field Theory and the Boundary Perspective

This analysis has several significant consequences:

• The operational selection of vacuum via UDW coupling underscores the boundary-centric, rather than time-evolution-centric, formulation of quantum field dynamics in the GBF.
• The necessity of importing external conditions to fix the vacuum for evanescent modes demonstrates intrinsic nonlocality: physical predictions inside a region $M$ depend on the embedding of $M$ in the larger spacetime.
• The approach offers a method for constructing a well-defined quantum field theory on arbitrary spacetime regions with boundaries, relevant for both theoretical investigations (e.g., in topological QFT or quantum gravity models) and for scenarios involving inhomogeneous materials or spacetime interfaces.
• This framework generalizes the notion of the Unruh effect and the detector-based definition of particle content to settings where standard global symmetry is absent or boundaries preclude a canonical choice.

7. Conclusion

Unruh-DeWitt type coupling provides a mathematically precise and operationally meaningful means of probing the vacuum structure of quantum scalar fields in boundary-sensitive contexts. In the General Boundary Formulation, the UDW detector operationalizes the selection of the field’s complex structure (and hence the vacuum) through its transition (or "click"/"no-click") probability. For propagating modes, this enforces the canonical vacuum uniquely; for evanescent modes, it exposes the essential nonlocality of the vacuum—its dependence on regions beyond the immediate spacetime domain. Thus, the UDW coupling in the GBF is not merely a calculational tool but provides a foundational apparatus for defining and probing quantum field theory on arbitrary spacetime domains (Banisch et al., 2012).