The Unruh–DeWitt detector and the vacuum in the general boundary formalism^*

To introduce the reader to the GBF of quantum theory we consider the quantum harmonic oscillator formulated in the language of the GBF in the following section. Then we consider holomorphic boundary states and amplitudes. We call this the boundary formulation of the quantum harmonic oscillator skipping the word 'general' since in the corresponding set up there is only one possible choice of boundary.

We then briefly review the case of a scalar general boundary QFT in section 3 in the holomorphic quantization scheme, discussing in particular the notion of unitarity in the theory, and the construction of one-particle-states and the one point function. The latter allows us to give the GBF formulation of the Unruh–DeWitt detector by using the text book formulas for the detector coupling. An independent derivation that happens fully in the GBF framework is given in appendix A.2, using coherent states which are reviewed in appendix A.1.

In section 4.1, we show that, using symmetry arguments and an appropriate choice of boundary states, we can determine the complex structure for propagating modes by requiring a detector at rest to not click. In section 4.2, we then consider an actual physical example in which evanescent waves appear as the appropriate choice of boundary state is unclear for evanescent waves using the GBF. This example is a toy model of a free electromagnetic field in the presence of a dielectric boundary. The excitation rates induced by evanescent waves in this case have been studied theoretically and experimentally in [CM71, CMD72]. We show that in order to reproduce the transition rates while maintaining the ability to model the two situations of particle emission and particle absorption as orthogonal subsets in the boundary Hilbert space, which is the situation one has in canonical quantization, the complex structure has to depend on the physical situation at the dielectric boundary, even if this boundary is outside of the spacetime region considered. We conclude that a complex structure for evanescent waves cannot be chosen by general physical arguments pertaining only to the spacetime region under consideration. We further show that the correct result for the transition rate of the detector is reproduced if a particular non-unitary complex structure is considered that reproduces the right asymptotic behavior. Such a non-unitary complex structure is used for the first time in interacting GBQFT in this paper.

While the standard formulation of quantum mechanics is concerned with the evolution of a system in the Hilbert space as it changes through time t, the GBF of quantum mechanics considers amplitudes between initial and final states. For zero-dimensional systems this is not more general, merely a rewriting that suggests an obvious generalization for the higher dimensional case.

To make this concrete for the zero-dimensional case we introduce a copy of the Hilbert space of the harmonic oscillator at each time t. These spaces are all canonically identified with the space at some fiducial time t = 0, $\mathcal {H}_t := \mathcal {H}_0$. We have the standard unitary evolution from $\mathcal {H}_t$ to $\mathcal {H}_{t^{\prime }}$ given by U(t', t). The usual statements we want to make in the theory concern probabilities to observe the system in a subspace $\mathcal {S}_f \subset \mathcal {H}_{f} := \mathcal {H}_{t_f}$ given that it was prepared in a subspace $\mathcal {S}_i \subset \mathcal {H}_{i} := \mathcal {H}_{t_i}$. We will denote the projectors on the subspace $\mathcal {S}_{i/f}$ by P_i/f. We introduce an orthonormal basis of $\mathcal {S}_{i/f}$ as $\psi ^a_{i}$ and $\psi ^a_{f}$. We model this situation in the text book way by taking the system at time t_i to be given by the density matrix $\frac{P_i}{{\rm tr} P_i}$, then at time t_f it will be in the state

$$\begin{eqnarray*} &&\sigma = U(t_f, t_i) P_i U(t_i, t_f)/{\rm tr}(P_i). \end{eqnarray*}$$

The text book probability to observe the proposition P_f is then given by tr(σP_f) or more explicitly

$$\begin{equation} P(\mathcal {S}_f|\mathcal {S}_i) = \frac{{\rm tr}(P_i U(t_i, t_f) P_f U(t_f, t_i))}{{\rm tr}(P_i)} = \frac{\sum _{a,b} \big|{\psi _f^b}^\dagger U(t_f,t_i) \psi _i^a \big|^2}{\sum _a \big|\psi _i^a \big|^2}. \end{equation} \tag{ 1 }$$

The general boundary formalism takes this as its starting point. We take the boundary Hilbert space $\mathcal {H}_{\partial I}$ of a segment I = [t_i, t_f] to be the space $\mathcal {H}_i \otimes \mathcal {H}_f^*$, where * indicates the Hermitian dual. Then $\mathcal {S}_i \otimes \mathcal {S}_f^* = \mathcal {S}_\mathcal {J}$ is the subspace of events we are interested in, relative to the preparation subspace $\mathcal {S}_i \otimes \mathcal {H}_f^* = \mathcal {S}_\mathcal {I}$, which only encodes the information on how the system was prepared. Note that here we have $\mathcal {S}_\mathcal {J}\subset \mathcal {S}_\mathcal {I}$.

We want to express the above formula given the spaces $\mathcal {S}_{\mathcal {I}}$ and $\mathcal {S}_{\mathcal {J}}$ directly. To do so it is convenient to consider the unitary evolution U as a map from the boundary space to $\mathbb {C}$, that is, we introduce $\rho _I: \mathcal {H}_i \otimes \mathcal {H}_f^\star \rightarrow \mathbb {C}$ as the linear map defined by

$$\begin{equation} \rho _I(\psi _i \otimes \psi _f^\dagger ) = \psi _f^\dagger U(t_f,t_i) \psi _i \end{equation} \tag{ 2 }$$

ρ_I is simply U(t_f, t_i) seen as an element of $\mathcal {H}_i^\star \otimes \mathcal {H}_f = \mathcal {H}_{\partial I}^*$. The map ρ_I can then also be read as the inner product in $\mathcal {H}_{\partial I}$, of the normalized state $\psi = \psi _i \otimes \psi _f^\dagger$ with the state $U^\dagger = \rho _I^\dagger \in \mathcal {H}_{\partial I}$. That is,

$$\begin{equation} \rho _I(\psi ) = \langle \rho _I^\dagger | \psi \rangle _{\partial I} = \langle U^\dagger | \psi \rangle _{\partial I}. \end{equation} \tag{ 3 }$$

Viewed this way the norm square of the map ρ_I can be seen as the length of the projection of $\rho _I^\dagger$ on the subspace spanned by ψ, that is, writing the projector onto that subspace P_ψ we have that

$$\begin{eqnarray*} &&|\rho _I(\psi )|^2 = |P_\psi \rho _I^\dagger |^2. \end{eqnarray*}$$

This allows for an immediate generalization to arbitrary subspaces $\mathcal {S}$. Given a projector P onto $\mathcal {S}$ and an arbitrary orthonormal basis ψ_c of $\mathcal {S}$, the amplitude can thus be written as

$$\begin{eqnarray*} &&|\rho _I \circ P|^2 = |P \rho _I^\dagger |^2 = \sum _c |\rho _I(\psi _c)|^2. \end{eqnarray*}$$

It is then straightforward to see that equation (1) can be written as

$$\begin{equation} P_{{\rm standard}}(\mathcal {S}_f|\mathcal {S}_i) = P_{{\rm GBF}}(\mathcal {S}_{\mathcal {J}}|\mathcal {S}_{\mathcal {I}}) = \frac{|\rho _I \circ P_{\mathcal {J}}|^2}{|\rho _I \circ P_\mathcal {I}|^2}. \end{equation} \tag{ 4 }$$

The central assumption of the general boundary formalism is that the above formula remains valid for wide classes of boundaries, subspaces and amplitude maps ρ_I in higher dimensions.

2.1.1. Example: The harmonic oscillator

For the harmonic oscillator with classical variables q, p we have the usual Hilbert space $\mathcal {H}_0 = L^2(\mathbb {R},{\rm d}q)$. We write $\mathcal {H}_t = L^2(\mathbb {R},{\rm d}q_t)$. The usual operators are

$$\begin{equation} \hat{p} \psi (q)=-{\rm i}\frac{{\rm d}}{{\rm d}q}\psi (q), \quad \hat{q} \psi (q)=q\psi (q) \end{equation} \tag{ 5 }$$

and, using the shorthand $M = \sqrt{m\Omega }$

$$\begin{equation} \hat{a}:= \frac{1}{\sqrt{2}}\left( M \hat{q}+{\rm i} \frac{1}{M}\hat{p}\right),\quad \hat{a}^\dagger :=\frac{1}{\sqrt{2}}\left(M \hat{q} - {\rm i} \frac{1}{M}\hat{p}\right), \end{equation} \tag{ 6 }$$

with $H = \Omega (\hat{a}^\dagger \hat{a} + \frac{1}{2})$, and the evolution map U(t_f − t_i) = exp (i(t_f − t_i)H).

From this we immediately have the general boundary construction above with

$$\begin{eqnarray*} &&\rho _{I}(\psi _i \otimes \psi _f^\dagger ) = \psi _f^\dagger \exp ({\rm i} (t_f - t_i) H)\psi _i. \end{eqnarray*}$$

For us it will be interesting to see an alternative construction of the GBF ingredients though, given by holomorphic quantization. This can be seen as a special case of geometric quantization [Woo97] and was introduced for affine and linear general boundary field theory in [Oec12b, Oec12a].

In the usual description of the harmonic oscillator above, we represent the algebra of observables by acting on the space of functions on just one of the canonical variables, q. These can be seen as functions on phase space satisfying $\frac{{\rm d}\psi }{{\rm d}p} = 0$. We can instead work with a different set of functions on phase space that instead satisfy $(M^{-1}\frac{{\rm d}}{{\rm d}q} - {\rm i} M\frac{{\rm d}}{{\rm d}p}) \psi = 0$. Introducing Q = Mq and P = M⁻¹p, and the variable a = Q + iP = Mq + iM⁻¹p, these are exactly the holomorphic functions of a. These are the functions that satisfy $\frac{{\rm d}\psi }{{\rm d}\overline{a}} = 0$, where $\frac{{\rm d}}{{\rm d}\overline{a}}$ is the Wirtinger derivative $\frac{{\rm d}}{{\rm d}\overline{a}} = \frac{{\rm d}}{{\rm d}Q} - {\rm i} \frac{{\rm d}}{{\rm d}P}$.

This space of holomorphic functions on the complex line,

$$\begin{eqnarray*} &&L^2_{{\rm hol}}(\mathbb {C},d_{{\rm Gauss}}), \end{eqnarray*}$$

with inner product

$$\begin{equation} \langle \psi ,\psi ^{\prime }\rangle :=\int _{\mathbb {C}}\frac{{\rm d}a}{2\pi }\,{\rm e}^{-\frac{1}{2}|a|^2} \overline{\psi (a)}\psi ^{\prime }(a), \end{equation} \tag{ 7 }$$

is particularly well suited for the harmonic oscillator and the bosonic scalar field.

The representation of the algebra of observables is easily given in terms of creation and annihilation operators $\hat{a}^\dagger$ and $\hat{a}$. On vectors $\psi (a) \in L^2_{{\rm hol}}(\mathbb {C})$ these are defined simply as multiplication and derivative operators respectively,

$$\begin{equation} \hat{a}^\dagger \psi (a) = \frac{a}{\sqrt{2}} \psi (a), \quad \hat{a} \psi (a) = \sqrt{2} \frac{{\rm d}}{{\rm d}a}\psi (a), \end{equation} \tag{ 8 }$$

where $\frac{{\rm d}}{{\rm d}a}$ is again a Wirtinger derivative $\frac{\rm d}{{\rm d}a} = \frac{\rm d}{{\rm d}Q} + {\rm i} \frac{\rm d}{{\rm d}P}$.

It is immediate that $\hat{a}$ and $\hat{a}^\dagger$ satisfy the commutation relations of the creation and annihilation operators. To check whether this is indeed a representation, we need to check that the Hermitian dual of $\hat{a}$ with respect to the Gaussian inner product above is indeed $\hat{a}^\dagger$, which follows from partial integration, and the fact that the Wirtinger derivative of the anti-holomorphic function $\overline{\psi }$ vanishes.

We can now also see how the holomorphic picture is well suited to the harmonic oscillator. The constant function ψ₀(a) = 1 is annihilated by $\hat{a}$ and thus gives the ground state. General states generated by $\hat{a}^\dagger$ and $\hat{a}$ are simply polynomials of a. Coherent states take the simple form,

$$\begin{eqnarray*} &&K_z(a) := \exp \left(\frac{1}{\sqrt{2}} z \hat{a}^\dagger \right) \psi _0(a) = \exp \left(\frac{1}{2}\overline{z} a\right) . \end{eqnarray*}$$

They are of particular interest because they fulfil the reproducing kernel property

$$\begin{equation} \langle K_z,\psi \rangle =\psi (z)\quad \mbox{\rm for any}\,\psi (a)\in L^2_{\rm hol}(\mathbb {C}) . \end{equation} \tag{ 9 }$$

This can be seen by expressing ψ(a) in a Taylor series, expressing the powers of a with powers of the creation operator $\hat{a}^\dagger$ and the vacuum state, using the duality of $\hat{a}$ and $\hat{a}^\dagger$, shifting the integration variable a to a − z and using that 〈ψ₀, ψ₀〉 = 1.

2.2.1. Complex structures

The space of holomorphic functions on phase space that is used depends on the choice of complex structure on phase space which provides an identification of phase space with the complex numbers. Different complex structures will lead to different function spaces.

This ambiguity is of no big consequence in finite systems. The Stone–von Neumann theorem ensures that all representations of the canonical commutation relations are unitarily equivalent, even though this might not be obvious in practice. This will change in the case of the scalar field that we will consider later in this paper, where the choice of complex structure can be shown to correspond to a choice of vacuum state. For this reason we will here discuss several different ways in which this choice of complex structure can be encoded.

The classical data which we have to start with is L, the phase space of our system, parametrized by canonically conjugate coordinates p,q, and equipped with an antisymmetric bilinear form ω, the Poisson bracket. We call the complexification of this phase space, $L^\mathbb {C}$.

The identification with the complex numbers can be encoded by specifying how the multiplication with the imaginary unit acts on phase space. That is, if π is the real-linear map from $\mathbb {C}$ to L, we are interested in the operator J defined by

$$\begin{eqnarray*} &&J\circ \pi = \pi \circ {\rm i}. \end{eqnarray*}$$

This operator is called a complex structure and has the property the J² = −1. Thus it has eigenvalues ±i. We can diagonalize it by lifting it to the complexified phase space $L^\mathbb {C}$. Its eigenspaces P_± are complex linear subspaces of $L^\mathbb {C}$ of complex dimension 1, as $P_+ = \overline{P_-}$. The projection on its +i eigenspace P₊,

$$\begin{eqnarray*} &&\pi ^{-1} = \case{1}{2} (1 - {\rm i} J), \end{eqnarray*}$$

provides us with an immediate identification of the real subspace L and $P_+ \simeq \mathbb {C}$. Its inverse is always just given by π( · ) = ℜ( · ). Thus a complex structure does indeed provide us with an identification of phase space with $\mathbb {C}$, as claimed. This is illustrated in figure 1.

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If the complex structure satisfies ω(J ·, J · ) = ω( ·, ·), and the symmetric linear form g( ·, ·) = ω(J ·, ·) is positive, we call J compatible with ω. The triple (ω, J, g) is called a Kähler structure on phase space. We will always require that ω and J define such a Kähler structure. This ensures that the inner product on $P_+ \subset L^\mathbb {C}$, defined in terms of g and ω as { ·, ·} = g(π( · ), π( · )) + 2iω(π( · ), π( · )), is Hermitian.

A more thorough and precise introduction to these structures in the general case will be given in section 3.

2.2.2. Harmonic oscillator continued...

We will now continue looking at the example of the holomorphic harmonic oscillator from section 2.1.1 and write down the above structures in this case.

We have $a = M q + \frac{i}{M}p$ so, introducing the shorthand notation X = (p, q)^T, the map π from L to $\mathbb {C}$ is explicitly given by $\pi ^{-1}(\mathbf {X}) = Mq + \frac{i}{M}p$. Then J is the matrix

$$\begin{eqnarray*} J = \left({\begin{array}{@{}cc@{}}0 & -1/M^2 \\ M^2 & 0\end{array}}\right). \end{eqnarray*}$$

The standard symplectic structure inherited by the dynamics is in these coordinates

$$\begin{eqnarray*} \omega (\mathbf {X}^{\prime },\mathbf {X}) = \frac{1}{2}(pq^{\prime } - qp^{\prime }) = (q^{\prime },p^{\prime }) \left({\begin{array}{@{}cc@{}}0 & 1/2 \\ -1/2 & 0\end{array}}\right)\left({\begin{array}{@{}c@{}}q \\ p\end{array}}\right). \end{eqnarray*}$$

It is straightforward to see that ω(J ·, J · ) = ω( ·, ·), and that the bilinear form

$$\begin{equation} g(\mathbf {X}^{\prime },\mathbf {X}) := 2\omega (\mathbf {X}^{\prime },J\mathbf {X}) = (q^{\prime },p^{\prime })^T \left({\begin{array}{@{}cc@{}}M^2 & 0 \\ 0 & 1/M^2\end{array}}\right) {{q}\choose{p}} \end{equation} \tag{ 10 }$$

is positive and symmetric and thus defines a natural metric on phase space. Thus ω, J and g form a Kähler structure as required. As noted above they define a complex inner product { ·, ·} via

$$\begin{eqnarray*} &&\lbrace \mathbf {X}^{\prime },\mathbf {X}\rbrace := g(\mathbf {X}^{\prime },\mathbf {X}) + 2{\rm i}\omega (\mathbf {X}^{\prime },\mathbf {X}). \end{eqnarray*}$$

This complex inner product is sesquilinear in the sense that {X', JX} = i{X', X} = −{JX', X}. In this case it is

$$\begin{eqnarray*} &&\lbrace \mathbf {X}^{\prime },\mathbf {X}\rbrace = \left(Mq^{\prime }-\frac{{\rm i}}{M}p^{\prime }\right)\left(Mq+\frac{{\rm i}}{M}p\right) = \bar{a}^{\prime }a, \end{eqnarray*}$$

the familiar sesquilinear form on $\mathbb {C}$. The eigenspaces of J in $L^\mathbb {C}\simeq \mathbb {C}^2$ are given by

$$\begin{equation*} P_\pm = \left\lbrace \left({\begin{array}{@{}cc@{}}z \\ \mp {\rm i}M^2 z\end{array}}\right), z \in \mathbb {C}\right\rbrace . \end{equation*}$$

More generally, the most general linear map on L which satisfies J² = −1 can be written as

$$\begin{equation} J = \left({\begin{array}{@{}cc@{}}b & -c^{-1}(1+b^2)^{\frac{1}{2}} \\ c (1+b^2)^{-\frac{1}{2}} & -b\end{array}}\right). \end{equation} \tag{ 11 }$$

It can be shown that J and ω are compatible, or equivalently that g( ·, ·) = 2ω( ·, J · ) is symmetric, and that g is positive, iff we have $(b,c) \in \mathbb {R}\times (0,\infty )$. The choice we made at the beginning of this section corresponds to b = 0 and c = M².

The eigenspace P₊ for the complex structure (11) can be written as

$$\begin{equation*} P_+ = \left\lbrace \left({\begin{array}{@{}c@{}}z \\ {\rm e}^{-{\rm i}\phi } c z\end{array}}\right), z \in \mathbb {C}\right\rbrace \end{equation*}$$

where e^−iϕ = (b − i)(ab² + 1)^−1/2, in other words ϕ ∈ [0, π].

Different complex structures lead to different identifications of phase space with the complex numbers. Thus the spaces of holomorphic functions for different complex structures, when viewed as functions on phase space, differ. However, it is still possible to compare functions in these different function spaces, by considering them as square integrable functions on phase space and taking an inner product there. This inner product is called a pairing [Woo81, Woo97].

Concretely, given two different identifications, a(p, q) = π⁻¹(p, q) ≠ a'(p, q) = π'⁻¹(p, q), and states ψ(a), ψ'(a) in $L^2_{{\rm hol}}(\mathbb {C},d_{{\rm Gauss}}a)$ and $L^2_{{\rm hol}}(\mathbb {C},d_{{\rm Gauss}}a^{\prime })$ respectively, we can construct functions $\phi (p,q) = {\rm e}^{-\frac{1}{4}|a(p,q)|^2} \psi (a(p,q))$, and $\phi ^{\prime }(p,q) = {\rm e}^{-\frac{1}{4}|a^{\prime }(p,q)|^2} \psi ^{\prime }(a^{\prime }(p,q))$ that are square integrable on phase space. That is, ϕ, ϕ' ∈ L²(L, dq dp).³

We denote the pairing between ψ and ψ' by (ψ, ψ')_p. Explicitly it is given by

$$\begin{equation} (\psi ,\psi ^{\prime })_p = \int {\rm d}q\, {\rm d}p \,{\rm e}^{-\frac{1}{4}|a(q,p)|^2-\frac{1}{4}|a^{\prime }(q,p)|^2} \overline{\psi (a(q,p))} \psi ^{\prime }(a^{\prime }(q,p)). \end{equation} \tag{ 12 }$$

This pairing has different physical interpretations that need to be established on a case by case basis.

2.3.1. Pairings for the harmonic oscillator

Note that as any classical time evolution T(t, t') generated by a Hamiltonian vector field leaves the symplectic form ω invariant, the operator J' = T(t, t')○J○T(t, t')⁻¹ is also a compatible complex structure. In this case the classical time evolution operator T(t, t'), considered as a map between the complex spaces arising from J and J', is unitary. As the Hermitian inner product on these complex spaces is defined in terms of ω and J, this will always be the case if we have J'○T(t, t') = T(t, t')○J.

Remarkably, for the harmonic oscillator the pairing between the holomorphic Hilbert spaces at different times provides us with the full quantum dynamics. To see this we can view phase space as the space of solutions x(t) to the equations of motion,

$$\begin{equation} {x}(t)= \frac{1}{2 M}(a_0 \,{\rm e}^{-{\rm i}\Omega t}+\overline{a_0} \,{\rm e}^{{\rm i}\Omega t}) . \end{equation} \tag{ 13 }$$

with

$$\begin{eqnarray*} &&a_0 = M q + {\rm i}\frac{p}{M} , \end{eqnarray*}$$

given in terms of our previous phase space coordinates. We then have an entire family of phase space coordinates,

$$\begin{eqnarray*} &&q_t = x(t) \;\;{\rm and} \;\; p_t = m \frac{{\rm d}x}{{\rm d} t} (t), \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&a_t = M q_t + {\rm i}\frac{p_t}{M} = \pi ^{-1}_t(q,p), \end{eqnarray*}$$

provides a whole family of complex structures on phase space. Note that the operators J can be written in very compact form as

$$\begin{equation} J = \frac{\partial _t}{\Omega }. \end{equation} \tag{ 14 }$$

Expressing this operator in the various coordinates q_t and p_t provides the complex structures a_t.

These complex structures are related quite simply by $a_t = {\rm e}^{{\rm i} \Omega (t - t^{\prime })} a_{t^{\prime }}$. As anticipated it is a unitary transformation. The pairing is now of the form

$$\begin{equation} (\psi _t,\psi _{t^{\prime }})_{{\rm pairing}} = \int {\rm d}q \,{\rm d}p \,{\rm e}^{-\frac{1}{4} |a_t|^2 - \frac{1}{4} |a_{t^{\prime }}|^2} \overline{\psi _t(a_t)} \psi _{t^{\prime }}(a_{t^{\prime }}), \end{equation} \tag{ 15 }$$

which, noting that $|a_t|^2 = |a_{t^{\prime }}|^2$ are real and thus equal to g² = g((q, p), (q, p)), we can simplify to

$$\begin{equation} (\psi _t,\psi _{t^{\prime }})_{{\rm pairing}} = \int {\rm d}q \, {\rm d}p \,{\rm e}^{-\frac{1}{2} g^2} \overline{\psi _t({\rm e}^{{\rm i} \Omega t}a_0)} \psi _{t^{\prime }}({\rm e}^{{\rm i} \Omega t^{\prime }}a_{0}). \end{equation} \tag{ 16 }$$

We can now check that we actually have

$$\begin{equation} (\psi ,\psi ^{\prime })_{{\rm pairing}} = \langle \psi | U(t - t^{\prime }) \psi ^{\prime } \rangle , \end{equation} \tag{ 17 }$$

where 〈|〉 is the inner product in $L^2_{{\rm hol}}(\mathbb {C},d_{{\rm Gauss}})$, with $U(t - t^{\prime }) = \exp (-{\rm i} \Omega \bar{a} a (t - t^{\prime }))$.

Thus, remarkably, the pairing between the Hilbert spaces associated to the different complex structures is related by the classical time translation and provides us with the full quantum dynamics. In general the pairing can be identified with semi-classical transition amplitudes. It is only for the harmonic oscillator and linear field theory, where it provides us with the Bogoliubov transformation, that the pairing provides the full quantum mechanical amplitudes.

The pairing can be evaluated very simply on coherent states, using the reproducing kernel property in equation (9) and the unitarity relation in equation (17) to obtain

$$\begin{equation} (K_{z,t},K_{z^{\prime },t^{\prime }})_{ {\rm pairing}} = {\rm e}^{\frac{1}{2} \overline{z^{\prime }} z}=K_{z^{\prime }}(z) . \end{equation} \tag{ 18 }$$

This ability to solve the dynamics in a straightforward way which makes the dependence on the underlying choices of complex structure obvious will also be true for the case of the field theory. This is what motivates our use of holomorphic quantization.

Note that the operator ordering in the quantum Hamiltonian is not the standard one for the harmonic oscillator. It differs by the ground state energy of $\frac{1}{2} \Omega$. This is the standard ordering for the free field theory, and the discrepancy for the harmonic oscillator will not matter for our application. The correct ordering can be obtained at the cost of significant technical complications, and we refer the interested reader to the literature, in particular the book [Woo97], on this point.

With the help of the pairing we can now give a very simple prescription for constructing the holomorphic general boundary harmonic oscillator for an interval I = [t_i, t_f] from the classical data.

The space of solutions for the harmonic oscillator is parametrized by the position and momentum at a certain time t, which, without adding any extra structure, we call $L_{t} \sim \mathbb {R}^2$. Given two copies of this phase space, one given by the parametrization L_i at time t_i and one by the parametrization L_f at time t_f we can form the direct sum of these spaces L_i⊕L_f to obtain the boundary phase space. The physical phase space L_p is then simply a two-dimensional subspace of this boundary phase space with elements (q_i, p_i, q_f, p_f) such that (q_i, p_i) ∈ L_i parametrizes the same solution as (q_f, p_f) ∈ L_f.

We can now directly quantize this setup using holomorphic quantization to obtain the GBF of the harmonic oscillator in this language. We put a complex structure on L_i, take the time translation related one at L_f, identify them with $\mathbb {C}$ and use the holomorphic Hilbert space construction to define the boundary Hilbert space for the region I = [t_i, t_f], that is, $\mathcal {H}_{\partial I} = \mathcal {H}_i \otimes \mathcal {H}_f^*$ with $\mathcal {H}_{i/f} = L_{{\rm hol}}$. We then use the pairing to define an amplitude map ρ_I

$$\begin{equation} \rho _I (\psi ^{\prime } \otimes \psi ^\dagger ) := (\psi ,\psi ^{\prime })_{ {\rm pairing}} = \int {\rm d}q \,{\rm d}p {\rm e}^{-\frac{1}{2}g^2} \overline{\psi (a_t)} \psi ^{\prime }(a_{t^{\prime }}) , \end{equation} \tag{ 19 }$$

where g² = g((q, p), (q, p)).

The structure we considered above was for the zero-dimensional QFT. The prescription of general boundary field theory for higher dimensions is to consider as classical boundary spaces, the space of solutions in the neighborhood of the boundary. The question of dynamics is then coded in the question if a particular solution near the boundary can be extended to the bulk.

This immediately generalizes our discussion above, as our boundary consists of two parts, which each individually corresponds to solutions near the boundary that are extensible throughout time. Thus the question of dynamics is simply if the extension from both parts of the boundary defines the same solution. This is another way to describe the subspace L_p⊂L_f⊕L_i given above.

In what follows we will always have a boundary with two components. We will always take the space of classical solutions near each of these components to be the space of solutions near them that can be extended throughout space time, that is, phase space. Thus the space of solutions near the entire boundary is the direct sum of two copies of phase space.

To then move on to construct a quantum theory based on this classical data we will need to again specify complex structures and construct Hilbert spaces on the spaces L. We will now do so for the special case of the Klein–Gordon field.

To give the Klein–Gordon fuel in the holomorphic GBF we need to classical phase space structure first. For any region $M\subseteq \mathcal {M}$ we have the action integral

$$\begin{eqnarray*} &&S_M(\phi ) := \int _M \sqrt{-g} L(\phi (x),\partial \phi (x)) \,{\rm d}x \end{eqnarray*}$$

where L(ϕ, ∂ϕ) is the usual Lagrangian density for the Klein–Gordon field on a Lorentzian spacetime

$$\begin{equation} L(\phi ,\partial \phi ,x):=\case{1}{2}g^{\mu \nu }\partial _\mu \phi (x)\partial _\nu \phi (x)-\case{1}{2} m^{2}\phi ^2(x) , \end{equation} \tag{ 20 }$$

As noted before, the GBF uses the real vector space L_Σ of solutions to the Euler–Lagrange-equations in a neighborhood of Σ. The dynamics can than be phrased as the problem of extending the solutions from a neighborhood of Σ to M.

The symplectic form can be derived from the action and is given by

$$\begin{eqnarray} &&\fl \omega _{\Sigma }: L_{\Sigma }\times L_{\Sigma }\rightarrow \mathbb {R}, \quad \omega _{\Sigma }(\phi ,\phi ^{\prime }) = \frac{1}{2}\int _{\Sigma } \sqrt{|\det g(x)|} n^\mu (x) \left(\phi ^{\prime }(x) \frac{\partial L}{\partial (\partial _\mu \phi )}(x) \right. \nonumber \\ &&\left. -\, \phi (x) \frac{\partial L}{\partial (\partial _\mu \phi ^{\prime })}(x) \right), \end{eqnarray} \tag{ 21 }$$

where n is the outward unit normal vector field to the hypersurface Σ.

If Σ = Σ₁∪Σ₂ as in (A2), we have $L_{\Sigma } = L_{\Sigma _1}\oplus L_{\Sigma _2}$, and $\omega _{\Sigma } = \omega _{\Sigma _1} + \omega _{\Sigma _2}$. Note that for $\overline{\Sigma }$, that is Σ with the opposite orientation we have $\omega _{\overline{\Sigma }} = - \omega _{\Sigma }$. For details of this construction, and proofs of the preceding statements, we refer again to the literature [Woo97].

In general the symplectic form does not need to be non-degenerate. Per [Woo97] this degeneracy is related to the presence of gauge orbits, or equivalently, the nonuniqueness of the extension of solutions from the neighborhood of a surface to the entire space time. Due to our assumption A3 this is not the case here, and we will assume non-degeneracy from here on.

By (A3), we obtain from $\omega _{\Sigma _{1/2}}$ a symplectic structure $\omega _{M_{1/2}}$ on the space L_M of global solutions on M. Furthermore, we assume that $\omega _{M_1}$ and $\omega _{M_2}$ coincide and just call both ω_M. This means that L_Σ ≃ L_M⊕L_M, and

$$\begin{eqnarray*} &&\omega _\Sigma ((\phi _1,\phi _2),(\phi ^{\prime }_1,\phi ^{\prime }_2)) = \omega _M(\phi _1,\phi _1^{\prime }) - \omega _M(\phi _2,\phi ^{\prime }_2), \end{eqnarray*}$$

where ϕ_1/2 ∈ L_M. This is of course just the case we described for the Harmonic oscillator.

Note that if we have an element $(\phi _1,\phi _2)\in L_\Sigma$ that extends to a solution on the interior of M we have that ϕ₁ and ϕ₂ denote the same element of L_M, and thus elements ϕ, ϕ' ∈ L_Σ that extend to the interior of M have ω_Σ(ϕ, ϕ') = 0. Therefore L_M forms a Lagrangian subspace of L_Σ. This property is actually not specific to our case, but true more generally, we refer the interested reader to the literature [Oec12b].

We now pick a polarization $\mathcal {P}$ of $L_{\Sigma }^{\mathbb {C}}$, that is a Lagrangian subset (the 'subset of positive frequency modes') $\mathcal {P} \subset L_{\Sigma }^{\mathbb {C}}$ of the complexification of L_Σ. If $\mathcal {P}$ is a polarization, then so is $\overline{\mathcal {P}}$, where the bar denotes complex conjugation in the usual sense. $\mathcal {P}$ is called a Kähler polarization if $\mathcal {P}\cap \overline{\mathcal {P}} = \lbrace 0\rbrace$. In this case, we can write $L_{\Sigma }^{\mathbb {C}} = \mathcal {P} \oplus \overline{\mathcal {P}}$.

Kähler polarizations are in one-to-one correspondence with complex structures J_Σ compatible to ω_Σ by

$$\begin{eqnarray*} \mathcal {P} & := & \big\lbrace \phi \in L_{\Sigma }^{\mathbb {C}} \big| J_{\Sigma }\phi = {\rm i}\phi \big\rbrace \\ \overline{\mathcal {P}} & := & \big\lbrace \phi \in L_{\Sigma }^{\mathbb {C}} \big| J_{\Sigma }\phi = -{\rm i}\phi \big\rbrace \end{eqnarray*}$$

For further details see [EMRV99]. We further always assume $J_\Sigma$ to be a positive compatible complex structure.

As in the case of the harmonic oscillator the sesquilinear form

$$\begin{eqnarray*} &&\lbrace \phi ,\eta \rbrace _{\Sigma } := g_{\Sigma }(\phi ,\eta ) + 2{\rm i}\omega (\phi ,\eta )\quad \forall \phi ,\eta \in L_{\Sigma } \end{eqnarray*}$$

turns the real vector space L_Σ into a complex Hilbert space, where multiplication with i is given by applying J_Σ. Real linear maps will lift to complex linear maps exactly if they commute with J_Σ. We have a canonical isomorphism $\pi : \mathcal {P} \rightarrow L_{\Sigma }$, $\pi (\psi ) = \psi + \bar{\psi }$ that identifies $\mathcal {P}$ and L_Σ, the latter now regarded as a complex vector space. The inverse is explicitly given in terms of the complex structure as

$$\begin{eqnarray*} &&\pi ^{-1} = \case{1}{2} (1 - {\rm i} J_\Sigma ). \end{eqnarray*}$$

In the case of the scalar field L_Σ as a complex space can be interpreted as the (dual of the) one-particle Hilbert space. This is analogous to the situation of the harmonic oscillator, there the complex classical phase space is simply $\mathbb {C}$, and can be identified with the one-dimensional space of first excitations.

The state space $\mathcal {H}_{\Sigma }^h$ of the holomorphic quantization is now constructed as⁴ $\mathcal {H}_{\Sigma }^h := L^2_{\rm hol}(L_{\Sigma },{\rm d}\mu _{\Sigma })$ the space of square-integrable holomorphic functions on L_Σ with respect to the inner product

$$\begin{eqnarray*} &&\langle \psi ^{\prime },\psi \rangle _{\Sigma } :=\int _{L_{\Sigma }}\psi (\phi )\overline{\psi ^{\prime }}(\phi ) {\rm d}\nu (\phi ) , \end{eqnarray*}$$

where dν is a Gaussian measure determined by $\frac{1}{2}g_{\Sigma }(\cdot ,\cdot )$ [Oec12b]. Given the state $\psi \in \mathcal {H}_{\partial M}$, the subspace L_M⊂L_∂M of global solutions on M and dν_M another Gaussian measure determined by $\frac{1}{4}g_{\partial M}(\cdot ,\cdot )$ [Oec12b], the amplitude ρ_M for a region M is given by

$$\begin{eqnarray*} &&\rho _M(\psi ) = \int _{L_{ M}} \psi (\phi ) {\rm d}\nu _M(\phi ) . \end{eqnarray*}$$

As anticipated in the harmonic oscillator section, in our case this is simply the pairing which provides us with a Bogoliubov transformation.

In this section, we briefly give the Fock space structure of $\mathcal {H}_{\Sigma }$, the state space of holomorphic quantization. Creation and annihilation operators and the one-particle Hilbert space are constructed, and the one point function is evaluated.

For an element $\xi \in L_\Sigma$ define the state $p_\xi \in \mathcal {H}_\Sigma$ by

$$\begin{equation} p_\xi (\phi ):=\frac{1}{\sqrt{2}}\lbrace \xi ,\phi \rbrace _\Sigma . \end{equation} \tag{ 22 }$$

We call p_ξ the normalized one-particle state corresponding to ξ ∈ L_Σ, for reasons to be clarified in the following. In [Oec12c], the actions of creation and annihilation operators on $\mathcal {H}^h_{\Sigma }$ have been worked out as

$$\begin{eqnarray} (a^\dagger _\xi \psi )(\phi ) & = & p_\xi (\phi )\psi (\phi )= \frac{1}{\sqrt{2}}\lbrace \xi ,\phi \rbrace _\Sigma \psi (\phi ) \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} (a_\xi \psi )(\phi ) & = & \langle K_{\Sigma ;\phi }, a_\xi \psi \rangle _\Sigma = \langle a^\dagger _\xi K_{\Sigma ;\phi }, \psi \rangle _\Sigma \ , \end{eqnarray} \tag{ 24 }$$

where K_{Σ; ϕ} is a coherent state around the classical solution ϕ, for details see appendix A.1.

Here, ξ ∈ L_Σ, and we may say that $a^\dagger _{\xi }$ and a_ξ create and annihilate 'a particle of type ξ', respectively. The action of a_ξ on coherent states is particularly simple:

$$\begin{eqnarray*} &&a_\xi K_{\Sigma ;\phi } = \frac{1}{\sqrt{2}}\lbrace \phi ,\xi \rbrace _\Sigma K_{\Sigma ;\phi }.\end{eqnarray*}$$

In particular, all annihilation operators annihilate the vacuum ψ_{0; Σ}(ϕ) = K_{Σ; 0}(ϕ) = 1. These operators satisfy the usual commutation relations

$$\begin{equation} [a_\xi ,a_\eta ^\dagger ]=\lbrace \eta ,\xi \rbrace _\Sigma ,\qquad [a_\xi ,a_\eta ]=0,\qquad [a_\xi ^\dagger ,a_\eta ^\dagger ]=0 . \end{equation} \tag{ 25 }$$

The creation and annihilation operators together with the vacuum state determine a Fock structure on $\mathcal {H}^h_{\Sigma }$. In particular, the one particle sector is the set

$$\begin{equation} \left\lbrace \frac{1}{\sqrt{2}}\lbrace \eta ,\cdot \rbrace ,\eta \in L_{\Sigma }\right\rbrace = \lbrace p_{\eta }, \eta \in L_{\Sigma }\rbrace =: L_{\Sigma }^* \simeq L_{\Sigma } , \end{equation} \tag{ 26 }$$

and we were indeed justified to call $p_\xi (\phi )=\frac{1}{\sqrt{2}}\lbrace \xi ,\phi \rbrace _\Sigma$ a one-particle-state. This is completely analogous to the harmonic oscillator case, where the space of first excitations is given simply by p_a( · ) = {a, ·}, with $a \in \mathbb {C}\sim L$.

To evaluate the one point function in the one particle sector we make use of the construction of observables in [Oec12d]. Let us introduce the short hand, $D(\phi )=\int {\rm d}x\,\sqrt{-\det g(x)}\mu (x)\phi (x)$, and the state η_D such that D( · ) = 2ω_∂M( ·, η_D). Then the amplitude in the presence of a source term μ can be evaluated on coherent states as:

$$\begin{eqnarray} \nonumber \rho ^{\mu }_M(K_{\xi })&=&\rho _M(K_\xi )\exp ({\rm i}D(\hat{\xi }(x))\,{\rm e}^{\frac{{\rm i}}{2}D(\eta _D)-\frac{1}{2}g_{\partial M}(\eta _D,\eta _D)}\\ &=&\rho _M(K_\xi )\exp ({\rm i}D(\hat{\xi }(x))\,{\rm e}^{\frac{{\rm i}}{2}D((1-{\rm i}J_{\partial M})\eta _D)}, \end{eqnarray} \tag{ 27 }$$

where $\hat{\xi }=\xi ^R-{\rm i}\xi ^I$ with ξ = ξ^R + J_∂Mξ^I and ξ^R, ξ^I ∈ L_M.

The one point function for the one particle sector is then simply obtained by taking the first order in ξ and the first derivative with respect to μ evaluated at μ = 0, which yields the simple form

$$\begin{equation} \rho ^{\phi (x)}(\lbrace \xi ,\cdot \rbrace ) = \hat{\xi }(x)\ . \end{equation} \tag{ 28 }$$

Let M be a region such that $\partial M = \Sigma _1 \cup \overline{\Sigma }_2$ is the disjoint union of two hypersurfaces. We can consider canonical projections $r_1: L_{ M} \rightarrow L_{\Sigma _1}$ and $r_2: L_{ M} \rightarrow L_{\Sigma _2}$.⁵ These are linear maps. If we assume these maps are invertible, we have the composition $T:= r_2\circ r_1^{-1}: L_{\Sigma _1}\rightarrow L_{\Sigma _2}$ evolves solutions at Σ₁ into solutions at Σ₂, hence it can be seen as a generalized notion of time evolution. We call T unitary if

$$\begin{equation} J_{\Sigma _2}\circ T = T\circ J_{\Sigma _1}. \end{equation} \tag{ 29 }$$

Note again that this defines a notion of unitarity for the classical evolution, which is of course not a necessary prerequisite for the quantum evolution to be unitary. In fact generally the classical time evolution will not be unitary in the presence of any potential or interacting terms. In [Oec12b] it was shown that if T is unitary, then there is a unitary map $U:\mathcal {H}_{\Sigma _1}\rightarrow \mathcal {H}_{\Sigma _2}$, Ψ↦Ψ○T⁻¹. In particular we have

$$\begin{eqnarray*} &&UK_{\Sigma _1,\xi } = K_{\Sigma _2,T\xi }. \end{eqnarray*}$$

Obviously T0 = 0 since T is linear. So the vacuum state on Σ₁ will evolve into the vacuum state on Σ₂, allowing us to construct a vacuum state K_{(0, 0)} on Σ₁∪Σ₂ which is in this sense compatible with the dynamics. Since we can extend this construction to any hypersurface Σ, there will be a global vacuum state compatible with the dynamics if T is unitary.

Note further that if T is unitary, then U defined in terms of it maps the one particle Hilbert space to the one particle Hilbert space, thus this can only be the case if there is no scattering or particle creation going on between hypersurfaces.

In this case $\hat{\xi }=\xi ^R-{\rm i}\xi ^I$ that arises in the one point function is given explicitly as the complex solution

$$\begin{equation} \hat{\xi }=r_1^{-1}\left(\case{1}{2}(1+{\rm i}J_{\Sigma _1})\xi _1+\case{1}{2}(1-{\rm i}J_{\Sigma _1})T^{-1}\xi _2\right) \end{equation} \tag{ 30 }$$

for all $\xi =(\xi _1,\xi _2)\in L_{\partial M}=L_{\Sigma _1}\times L_{\Sigma _2}$.

More generally using the unitary map U the amplitude map can be expressed using the inner product in the form

$$\begin{equation} \rho _{M}(\Psi _1\otimes \iota _{\Sigma _2}{\Psi }_2)=\rho _{M}(\Psi _1\otimes \overline{{\Psi }_2})=\langle U^{-1}\Psi _2, \Psi _1\rangle _{\Sigma _1} \end{equation} \tag{ 31 }$$

for all $\Psi _1\in \mathcal {H}_{\Sigma _1}$ and $\Psi _2\in \mathcal {H}_{\Sigma _2}$ and $\iota _{\Sigma _2}$ given by

$$\begin{equation} \iota _\Sigma :\mathcal {H}_{\Sigma }\rightarrow \mathcal {H}_{\overline{\Sigma }} ,\quad \psi \mapsto \overline{\psi } . \end{equation} \tag{ 32 }$$

Let us assume that we are given some symmetries of the classical spacetime region the Klein–Gordon field under consideration is defined on. Of course, we would wish to find these symmetries also represented in the QFT. Imposing this as a condition can be used to reduce the ambiguity in the choice of complex structure considerably.

Let {χ_i} be a collection of vector fields encoding the classical spacetime symmetries, i.e. Killing vector fields. Then we demand that the complex structure $J_\Sigma$ acting on the field configurations ϕ at the hypersurface Σ commutes with the vector fields $\chi _i^\mu \partial _\mu$ as $[\chi _i^\mu \partial _\mu ,J_{\sigma }]\phi (x)|_{\Sigma }=0$ for all i. Let $L_\Sigma ^{\lbrace i\rbrace }\subseteq L_{\Sigma }$ be the eigenspace of $\chi _i^\mu \partial _\mu$ to the collection of eigenvalues {λ_i}. The complex structure J_Σ must then leave $L_\Sigma ^{\lbrace i\rbrace }$ invariant, i.e. $J_{\Sigma }L_\Sigma ^{\lbrace i\rbrace }=L_\Sigma ^{\lbrace i\rbrace }$ for all i. Hence, we can try to fix J_Σ for every i separately.

Let us for example consider the case of 1 + 1-dimensional Minkowski space covered by coordinates (t, x₁) in which the metric takes the form g = diag(1, −1). Let us assume that we are given a region M = Σ × [x_{1, l}, x_{1, r}] in that spacetime bounded by two hyperplanes Σ_l and Σ_r at x_{1, l} and x_{1, r} respectively with x_{1, l} < x_{1, r}. Then the symmetries of this region are clearly time translation, time reflection and space reflection at the hyperplane at (x_r − x_l)/2. Hence, imposing the compatibility with the time translation invariance on $J_{\Sigma _l}$ and $J_{\Sigma _r}$ means that they must commute with the vector field E = i∂_t. Imposing this condition we find that $J_{\Sigma _l}$ and $J_{\Sigma _r}$ must leave the energy eigenspaces $L^{E}_{\Sigma _l}\subset L_{\Sigma _l}$ and $L^{E}_{\Sigma _r}\subset L_{\Sigma _r}$ respectively invariant.

Thus the choice of complex structure reduces from one posed in an infinite dimensional vector space into one choice of J_E per L^E. As L^E is two dimensional, namely the modes exp (i(Et ± px)), the complex structures allowed per energy eigenspace are the same as those of the harmonic oscillator.

We will make use of that throughout sections 4.1 and 4.2.

We can now discuss the coupling of an Unruh–DeWitt type detector to the GBF states we have described above. In first order perturbation theory the response amplitude of the detector at a given field configuration is provided by the one point function of the field.

Consider a harmonic oscillator of reduced mass $\sqrt{m\Omega }$, and at coupling strength λ, traveling on a world line γ in the presence of a field transition from the vacuum to ψ. Th amplitude of its transition between states differing by frequency Ω can be deduced immediately from the text book treatment of Birrell and Davis [BD82] as

$$\begin{equation} \fl \rho _{BD} = {\rm i} \lambda \sqrt{\frac{2}{m\Omega }} \int {\rm d}\tau {\rm e}^{{\rm i} \Omega \tau } \langle \psi |\phi (\gamma (\tau ))|0\rangle = {\rm i} \lambda \sqrt{\frac{2}{m\Omega }} \int {\rm d}\tau {\rm e}^{{\rm i} \Omega \tau } \rho ^{\phi (\gamma (\tau ))}(|\psi \rangle \otimes |0\rangle ) . \end{equation} \tag{ 33 }$$

The discussion of perturbation theory carries over to the GBF case in much the same way as in the standard treatment. A derivation from coherent states is given in appendix A.2. One key difference, as discussed above is that we specify the state on the whole boundary, that is, the one particle states are no longer necessarily of the form |ψ〉⊗|0〉 but can be superpositions of those. Their general functional form is {ξ, ·}. Recall that in the presence of such a boundary state, we found the one point function to have the simple form

$$\begin{equation} \rho ^{\phi (x)}(\lbrace \xi ,\cdot \rbrace ) = \hat{\xi }(x). \end{equation} \tag{ 34 }$$

Inserting this into the standard result we find that the probability P = |ρ_BD|² from the ground state of the harmonic oscillator to the first excited state is given by

$$\begin{eqnarray} P^{\Omega ,\xi }(g\rightarrow e)&:=&\frac{2\lambda ^2}{m\Omega }\int _{\tau _1}^{\tau _2} {\rm d}\tau \, {\rm d}\tau ^{\prime }\, {\rm e}^{{\rm i}\Omega (\tau -\tau ^{\prime })} \hat{\xi }(\gamma (\tau ))\overline{\hat{\xi }(\gamma (\tau ^{\prime }))}. \end{eqnarray} \tag{ 35 }$$

For the corresponding deexcitation probability we have

$$\begin{eqnarray} P^{\Omega ,\xi }(e\rightarrow g)&:=&\frac{2\lambda ^2}{m\Omega }\int _{\tau _1}^{\tau _2} {\rm d}\tau \, {\rm d}\tau ^{\prime }\, {\rm e}^{-{\rm i}\Omega (\tau -\tau ^{\prime })} \hat{\xi }(\gamma (\tau ))\overline{\hat{\xi }(\gamma (\tau ^{\prime }))}. \end{eqnarray} \tag{ 36 }$$

Appendix A.2 contains a detailed derivation of these formulas, demonstrating that we are justified to use the standard detector formula for states specified on time like hyper surfaces.

Let us return for a moment to the interpretation of these probabilities in the presence of a boundary state {ξ, ·}. From the derivation in appendix A.2 we know that these probabilities are normalised to the one particle sector. Thus they express the joint probability that the harmonic oscillator experiences the described transition, and that we find the field in the particular one particle configuration ξ, relative to the probability that we find at most one particle on the boundary, which for the given situation turns out to be 1.

Recall that we always consider the case that the boundary of the region M is a disjoint union ∂M = Σ₁∪Σ₂. Thus we can write the vacuum state as ψ_{M; 0} = ψ_{1; 0}⊗ψ_{2; 0}. Then, we call ψ_{1; 0} the vacuum state on Σ₁ and ψ_{2; 0} the vacuum state on Σ₂, and we can consider a ξ such that $a^\dagger _\xi \psi _{M;0}=\psi _{1;0}\otimes \psi _{2}$ for some $\psi _2\in \mathcal {H}_{\Sigma _2}$.

With this particular configuration of the boundary field we recover the interpretation as a transition from the vacuum state on Σ₁ to a particular one particle state on Σ₂. If Σ₁ is in the past of Σ₂ the equations (35) and (36) have the interpretation of the probability that the detector transitions and that the field goes from the vacuum to the state ψ₂.

In order to study the probability that the detector transitions given just the fact that the field was in the vacuum state at Σ₁ we would need to sum (35) or (36) over an orthonormal basis of one particle states of the form ψ_{1; 0}⊗ψ₂, as per the general form of the probability interpretation.

The central feature that makes the particular probabilities (35) and (36) so useful for our study is that as $\hat{\xi }$ depends very explicitly on the complex structure. For example, where applicable equation (30) expresses $\hat{\xi }$ explicitly in terms of J. This will make it straightforward to study the dependence of the transition probabilities on the choice of complex structure.

In this section we show how the detector response can be used to formulate a condition that determines the complex structure for propagating waves when we claim that the complex structure may possess all the symmetries of the spacetime region. The idea is to require that all one particle states with outgoing energy flux have no probability of exciting the detector at rest.

We are dealing with 1 + 1-dimensional Minkowski space and a region M = Σ × [x_{1, l}, x_{1, r}] bounded by parallel timelike hyperplanes Σ_l and Σ_r where Σ_l is oriented as Σ pointing outside of the region M and Σ_r has opposite orientation. Hence, the symmetries in the global coordinates in which the time direction is tangential to the timelike hyperplanes and in which the metric takes the form g = diag(1, −1) at every point of M are time translation and time reversal. We use exactly this coordinate system in the following. The symplectic form is then given as

$$\begin{equation} \fl \omega _{\partial M}(\phi ,\phi ^{\prime })=-\frac{1}{2}\int _{-\infty }^{\infty } {\rm d}x_0 (\phi \partial _{x_1}\phi ^{\prime }-\phi ^{\prime }\partial _{x_1}\phi )|_{x_{1,l}}+\frac{1}{2}\int _{-\infty }^{\infty } {\rm d}x_0 (\phi \partial _{x_1}\phi ^{\prime }-\phi ^{\prime }\partial _{x_1}\phi )|_{x_{1,r}} . \end{equation} \tag{ 40 }$$

We assume the Unruh–DeWitt detector to be at rest, i.e. γ(τ) = (τ, z) inside the region M throughout this whole section (see figure 2).

The additional ingredient for determining the complex structure is the no-click condition for propagating waves which is the following requirement: we introduce the set of positive flux modes $L^{\mathbb {C}}_{\partial M,+}\subset L^{\mathbb {C}}_{\partial M}$ consisting of solutions $\xi _{p_0}={\rm e}^{-{\rm i}p_0 x_0}(a{\rm e}^{{\rm i}p_1x_1}+b{\rm e}^{-{\rm i}p_1x_1})$ fulfilling

$$\begin{equation} -n_\mu T^{\mu 0}({\mathfrak {R}e}(\xi _{p_0}))(x)>0 \end{equation} \tag{ 41 }$$

at every point x ∈ ∂M with T^μν the energy–momentum tensor of the field and n the unit normal covector field to ∂M pointing out of the region M. Now, to determine the complex structure we require that the set of states corresponding to positive flux modes should give a vanishing transition rate of the detector.

Comparison to the case of spacelike hypersurfaces

Let us briefly consider the condition that all one particle states with outgoing energy flux have no probability of exciting the detector at rest, which we formalized above, for the case of two spacelike hypersurfaces. In that case we have a priori four modes per energy subspace of the complexified state space at each spacelike hypersurface, two future pointing, and two past pointing.

Out of these the complex structure picks two per surface as +i eigenvalues, or, physically speaking, 'positive' energy modes. The condition that we consider the modes that flow out of the space time then chooses the past pointing modes for the past hypersurface and the future-pointing modes for the future hypersurface. On the other hand, the usual complex structure chooses the future pointing modes in both cases. Thus the intersection between the two are the positive frequency modes on the future hyperplane, and we are considering vacuum to one particle transitions, which, with the detector in the ground state, cannot happen.

Any other choice of modes for the complex structure would lead to the presence of modes on the past hypersurface, and thus allow for the excitation of the detector. Thus turning the above on its head, we can use the condition that the detector should not click for outward pointing modes to uniquely fix the complex structure in the case of spacelike hypersurfaces.

Note again that we have two separate notions of positivity here, one determined by the complex structure, and the other, independently of it, by the energy flux of a certain mode.

The condition that energy should be flowing out of the region can be seen simply as a generalized version of the condition that the past hypersurface should be in the vacuum state. In particular it is a notion that survives complexification and thus can be used, in combination with the detector response, to pick the vacuum state.

To make the no-click condition mathematically clear we first assume that we are dealing with a unitary complex structure as defined in section 3.3. Since the complex structure respects the symmetries of the classical configuration we can deal with all the energy subspaces of $L^{\mathbb {C}}_{\Sigma }$ separately. To determine the complex structure we look at the space $L^{\mathbb {C},E}_{\Sigma }:=L^{\mathbb {C},p_0}_{\Sigma }\oplus L^{\mathbb {C},-p_0}_{\Sigma }$ with $L^{\mathbb {C},p_0}_{\Sigma }=\lbrace \sum _{\sigma _1=\pm }a_{\sigma _1}e(p_0,\sigma _1p_1)|\,a_{\sigma _1}\in \mathbb {C}\rbrace$ where $e(p_0,p_1)={\rm e}^{-{\rm i}(p_0 x_0 - p_1 x_1)}$. Using this split we investigate the no-click condition for elements of $L^{\mathbb {C}}_{\partial M,+}\cap L^{\mathbb {C},E}_{\Sigma }$ with $L^{\mathbb {C}}_{\partial M,+}$ introduced above. We find that on Σ_l the condition in equation (41) becomes

$$\begin{equation} T^{10}({\mathfrak {R}e}(\xi _{p_0}))(x)= -(\partial _0 {\mathfrak {R}e}(\xi _{p_0}) ) (\partial _1 {\mathfrak {R}e}(\xi _{p_0}) )(x)>0, \end{equation} \tag{ 42 }$$

and with the opposite sign on Σ_r. We obtain that solutions in $L^{\mathbb {C}}_{\partial M,+}\cap L^{\mathbb {C},E}_{\Sigma }$ are of the form $\xi ^{\mathbb {C}}_{+}=(\xi _l,\xi _r)$ with ξ_l = c_{l, +}e(p₀, p₁) + c_{l, −}e( − p₀, −p₁) and ξ_r = c_{r, +}e(p₀, −p₁) + c_{r, −}e( − p₀, p₁). We define the boundary configuration to be $\xi _{+}:=\mathfrak {Re}(\xi ^{\mathbb {C}}_{+})\in L_{\partial M}$. Let the coefficients c_{i, ±} be given such that ξ₊ corresponds to a normalized state $a^{\dagger }_{\xi _{+}}\psi _{M;0}$, we find that for the field being in this state the probability for the transition of a detector at rest, i.e. worldline γ(τ) = (τ, z) from the ground state to the excited state following from equation (35) is given as

$$\begin{eqnarray} &&P_\eta (g\rightarrow e) = \frac{2\lambda ^2}{m\Omega }\int _{-\infty }^{\infty } {\rm d}\tau\, {\rm d}\tau ^{\prime }\, {\rm e}^{{\rm i}\Omega (\tau -\tau ^{\prime })} \hat{\xi }_{+}(\tau , z)\overline{\hat{\xi }_{+}(\tau ^{\prime },z)} . \end{eqnarray} \tag{ 43 }$$

Let us assume that the complex structure $J_{\partial M}=(J_{\Sigma _l},J_{\Sigma _r})$ is unitary (see section 3.3). Then, we have

$$\begin{eqnarray} &&\hat{\xi }_{+} = \xi _{+}^R-i\xi _{+}^I=\case{1}{2} (1+iJ_{\Sigma _l})\mathfrak {Re}(\xi _l)+\case{1}{2} (1-iJ_{\Sigma _l})\mathfrak {Re}(\xi _r) . \end{eqnarray} \tag{ 44 }$$

The no-click condition is required to hold for every element of $L^{\mathbb {C}}_{\partial M,+}$, that is for every set of coefficients c_{l, ±} and c_{r, ±}. Considering the special case of c_{l, +} = c_{l, −} and c_{r, +} = c_{r, −} = 0 we obtain that

$$\begin{eqnarray} &&\fl \hat{\xi }_{+}=\xi _{+}^R-{\rm i}\xi _{+}^I=c_{r,+}\left(\case{1}{2}(1+{\rm i}J_{\Sigma _l})e(p_0,p_1)+\case{1}{2}(1+{\rm i}J_{\Sigma _l})e(-p_0,-p_1)\right) . \end{eqnarray} \tag{ 45 }$$

which means that for the detector not to click we need $(1+{\rm i}J_{\Sigma _l})e(p_0,p_1)=0$ which leads to $J_{\Sigma _l}e(p_0,p_1)={\rm i}e(p_0,p_1)$ and with complex conjugation $J_{\Sigma _l}\overline{e(p_0,p_1)}=J_{\Sigma _l}e(-p_0,-p_1)=-{\rm i}e(-p_0,-p_1)$. Together with the time reversal invariance which tells us that $J_{\Sigma _l}e(-p_0,p_1)={\rm i}e(-p_0,p_1)$ and $J_{\Sigma _l}e(p_0,-p_1)=-{\rm i}e(p_0,-p_1)$ this already determines the complex structure to be

$$\begin{equation} J_{\Sigma _l}=-J_{\Sigma _r}=\frac{\partial _{x_1}}{\sqrt{-\partial ^2_{x_1}}}. \end{equation} \tag{ 46 }$$

Putting this back into equation (44) we find

$$\begin{eqnarray} &&\hat{\xi }_{+}=\case{1}{2}(\overline{c_{l,+}}+c_{l,-})e(-p_0,-p_1)+\case{1}{2}(\overline{c_{r,+}}+c_{r,-})e(-p_0,p_1) . \end{eqnarray} \tag{ 47 }$$

With (43) we obtain that the no-click condition is fulfilled independently of the choice of the coefficients c_{l, ±} and c_{r, ±} as it was required. In particular, this includes states that can be seen as consisting of a one particle state on the right and the vacuum state on the left hypersurface and vice versa. But the above considerations were much more general as we also considered general one particle states on the whole boundary ∂M.

For evanescent wave solutions the property to be in the set of positive flux modes fulfilling condition (41) depends on the position of the hypersurface, i.e. for two hypersurfaces Σ_ζ and $\Sigma _{\zeta ^{\prime }}$ at x₁ = ζ and x₁ = ζ' respectively the set of positive flux modes is a different subset of $L^{\mathbb {C}}_{\partial M}$. In the derivation of the complex structure via the no-click condition we used explicitly that the two sets of positive flux modes for Σ_l and $\overline{\Sigma _r}$ coincide. We conclude that the no-click condition in the form above is not applicable to evanescent waves. In the next section we will construct an explicit physical situation to show a way how to fix the complex structure for evanescent waves on timelike hypersurfaces using the Unruh–DeWitt detector and our knowledge about the complex structure on spacelike hypersurfaces.

An example of a physical situation in which evanescent waves can be prepared is Maxwell electrodynamics in the presence of a dielectric slab. This was investigated in [CM71] where the authors model the experimental situation as two half spaces of different refractive index with one of them being the vacuum refractive index. In this setup solutions exist that are propagating waves in the dielectric medium and become evanescent in the vacuum. The authors of [CM71] find after quantization that their theoretical results coincide with semi-classical calculations and in a later publication [CMD72] they also show that they coincide with experiments. We are going to consider a similar setup for the massive scalar field in 1 + 1 dimensions. This toy model recovers the basic situation found in [CM71] but makes it much easier to understand the calculations which will allow us to focus on the basic insights concerning evanescent waves. We will use the Unruh–DeWitt detector as a toy model for the atoms in the electromagnetic field near the dielectric boundary. Throughout this section we will again assume the detector to be at rest, i.e. γ(τ) = (τ, z).

The setup we consider is shown in figure 3. The 1 + 1-dimensional spacetime $\mathcal {M}$ covered by coordinates (x₀, x₁) is separated in three regions with different metric tensor g. For region I and III we have in coordinates

$$\begin{eqnarray*} g=\left(\begin{array}{@{}cc@{}}\frac{1}{n} & 0 \\ 0 & -1\end{array} \right) \end{eqnarray*}$$

and for region II we have:

$$\begin{eqnarray*} g=\left(\begin{array}{@{}cc@{}}1 & 0 \\ 0 & -1\end{array}\right). \end{eqnarray*}$$

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Zoom In Zoom Out Reset image size

The boundaries of the regions are at −ζ and ζ. We consider the action

$$\begin{equation} S[\phi ]=\frac{1}{2}\int \mathrm{d}^2x\,\sqrt{-g}(g^{\mu \nu }\partial _\mu \phi \partial _\nu \phi -m^2\phi ^2). \end{equation} \tag{ 48 }$$

for the free Klein–Gordon field. The conditions for the construction of solutions ϕ of the Klein–Gordon equation

$$\begin{equation} (\partial _\mu \sqrt{-g}\,g^{\mu \nu }\partial _\nu +\sqrt{-g}\,m^2)\phi (x)=0 , \end{equation} \tag{ 49 }$$

at −ζ and ζ can be derived from the action as follows: Let us assume that we are given a spacetime $\mathcal {M}$ and a timelike hypersurface Σ separating $\mathcal {M}$ in two regions M₁ and M₂. Let dσ_{i, μ} be the induced volume measure on Σ_i, and define Σ₁ = Σ and $\Sigma _2=\overline{\Sigma }$. By varying the action (48) by δϕ and partial integration we obtain

$$\begin{equation} \fl \delta S=-\sum _i\int _{M_i} \mathrm{d}^2x\,\delta \phi (\partial _\mu \sqrt{-g}\,g^{\mu \nu }\partial _\nu +\sqrt{-g}\,m^2)\phi (x)+\beta \sum _i\int {\rm d}\sigma _{i,\mu }\,\delta \phi \,g^{\mu \nu }\partial _\nu \phi |_{\Sigma _i} , \end{equation} \tag{ 50 }$$

where β depends on the orientation of Σ. The first term leads to the Euler–Lagrange equation (49) in M₁ and M₂ and the second terms leads to the condition that for ϕ continuous at Σ and a jump in g we must have a jump in the first derivative of ϕ at Σ. For the case of the dielectric slabs we obtain the boundary conditions

$$\begin{eqnarray} &&\nonumber \lim _{\epsilon \rightarrow 0}\frac{1}{\sqrt{n}}\partial _1\phi |_{-\zeta -\epsilon }=\lim _{\epsilon \rightarrow 0}\partial _1\phi |_{-\zeta +\epsilon }\quad {\rm and}\\ &&\lim _{\epsilon \rightarrow 0}\partial _1\phi |_{\zeta -\epsilon }=\lim _{\epsilon \rightarrow 0}\frac{1}{\sqrt{n}}\partial _1\phi |_{\zeta +\epsilon }. \end{eqnarray} \tag{ 51 }$$

With the separation

$$\begin{equation} \phi ^\pm _{p_0}(x)=\varphi ^\pm _{p_0}(x_1)\,{\rm e}^{-{\rm i}p_0x_0} \end{equation} \tag{ 52 }$$

with p₀ > 0 we obtain for $p_0^2<m^2$ the set of solutions $\varphi ^+_{p_0}(x_1)$ defined in the appendix A.3 in equation (A.24) and $\varphi ^-_{p_0}(x_1)=\varphi ^{+}_{p_0}(-x_1)$. These solutions are chosen such that with the symplectic form for an equal time hypersurface given as

$$\begin{equation*} \omega _{x_0}(\phi ,\phi ^{\prime })=\frac{1}{2}\int _{-\infty }^{\infty } {\rm d}x_1 \sqrt{-g} g^{00}(\phi \partial _0\phi ^{\prime }-\phi ^{\prime }\partial _t\phi )|_{x_0} . \end{equation*}$$

they fulfill the orthonormality relation with the Klein–Gordon inner product

$$\begin{equation} \big(\phi ^s_{p_0},\phi ^{s^{\prime }}_{p^{\prime }_0}\big)_{KG}:={\rm i}\omega _{x_0}\big(\overline{\phi ^{s}_{p_0}},\phi ^{s^{\prime }}_{p^{\prime }_0}\big)=2\pi \sqrt{n}p_0 \delta (p-p^{\prime })\delta _{s,s^{\prime }}. \end{equation} \tag{ 53 }$$

With equation (53) the canonical quantization prescription leads to the absorption and emission operator

$$\begin{equation*} \hat{\phi }(x)=\int _0^\infty \frac{{\rm d}p}{2\pi \sqrt{n}p_0}\sum _{s=\pm }\big(a_{p,s}\phi _{p_0}^s(x)+{\rm h.c.}\big) \end{equation*}$$

with the commutation relations $[a_{p,s}^\dagger ,a_{p^{\prime },s^{\prime }}]=(\phi ^s_{p_0},\phi ^{s^{\prime }}_{p^{\prime }_0})_{{\rm KG}}=2\pi \sqrt{n} p_0\delta (p-p^{\prime })\delta _{s,s^{\prime }}$. We define the momentum $p=(np_0^2-m^2)^{1/2}$ in region I and III and $p_1=(m^2-p_0^2)^{1/2}$ in region II, respectively. The total probabilities for an Unruh–DeWitt detector with the world line γ(τ) = (τ, z) to emit a field particle while passing from its ground state to an exited state and vice versa is then given as

$$\begin{eqnarray} &&\fl P^{\Omega }(g\rightarrow e)=\int _0^\infty \frac{{\rm d}p}{2\pi \sqrt{n}p_0}\sum _{s=\pm }\left|\langle \psi _0|a_{p,s}\frac{\lambda }{m\Omega }\int _{-\infty }^{\infty } \tau \hat{\phi }(\gamma (\tau )) |\psi _0\rangle \right|^2 \nonumber \\ &&{\hphantom{\fl P^{\Omega }(g\rightarrow e)}}=\frac{\lambda ^2}{m\Omega }\int _0^\infty \frac{{\rm d}p}{2\pi \sqrt{n} p_0}\int _{-\infty }^\infty {\rm d}\tau\, {\rm d}\tau ^{\prime } {\rm e}^{{\rm i}(\Omega +p_0)(\tau -\tau ^{\prime })}\sum _{s=\pm }\big|\varphi _{p_0}^{s}(z)\big|^2 \nonumber \\ &&\fl P^{\Omega }(e\rightarrow g)=\frac{\lambda ^2}{m\Omega }\int _0^\infty \frac{{\rm d}p}{2\pi \sqrt{n} p_0}\int _{-\infty }^\infty {\rm d}\tau\, {\rm d}\tau ^{\prime } {\rm e}^{-{\rm i}(\Omega -p_0)(\tau -\tau ^{\prime })}\sum _{s=\pm }\big|\varphi _{p_0}^{s}(z)\big|^2 . \end{eqnarray} \tag{ 54 }$$

It turns out that the probability P^Ω(g → e) vanishes and with the expression in equation (A.24) we find the transition probability

$$\begin{eqnarray} &&\fl P^{\Omega }(e\rightarrow g):=\frac{\lambda ^2 \sqrt{n}}{m\Omega p}\sum _{s=\pm }|\varphi _{\Omega }^{s}(z)|^2 \nonumber \\ &&{\hphantom{\fl P^{\Omega }(e\rightarrow g)}} = \frac{\lambda ^2 \sqrt{n}}{m\Omega p}|N_{\Omega }|^2\frac{p}{\sqrt{n}p_1}\left(\left(\frac{\sqrt{n}p_1}{p}-\frac{p}{\sqrt{n}p_1}\right)\right.\nonumber \\ &&\left.+\frac{1}{2}\left(\frac{\sqrt{n}p_1}{p}+\frac{p}{\sqrt{n}p_1}\right)\left(\cosh 2p_1(z + \zeta ) + \cosh 2p_1(z-\zeta )\right)\right)\delta (0)^2 \nonumber \\ &&{\hphantom{\fl P^{\Omega }(e\rightarrow g)}} =\frac{\lambda ^2 }{m\Omega p_1}|N_{\Omega }|^2\left(\left(\frac{\sqrt{n}p_1}{p}-\frac{p}{\sqrt{n}p_1}\right) \right. \nonumber \\ &&\left. +\left(\frac{\sqrt{n}p_1}{p}+\frac{p}{\sqrt{n}p_1}\right)\cosh 2p_1 z\,\cosh 2p_1\zeta \right)\delta (0)^2 , \end{eqnarray} \tag{ 55 }$$

that depends on the space point z at which the detector rests and on the energy Ω and we have the usual divergence δ(0)² due to the infinite duration of the interaction. If we want to know the probabilities for the absorption of a particle by the detector undergoing the same transitions as above we just have to interchange Ω by −Ω in equation (54). Then we find that $P_{{\rm absorption}}^{\Omega }(e\rightarrow g)=0$ and $P_{{\rm absorption}}^{\Omega }(g\rightarrow e)=P_{{\rm emission}}^{\Omega }(e\rightarrow g)$ where the latter is the transition probability in equation (54).

The GBF description

We now give a GBF description for a region in between the dielectric slabs. Take the region M = Σ × [x_{1, l}, x_{1, r}] with timelike boundaries Σ_l and Σ_r with unit normal pointing outside of M and M lying completely in region II (see figure 4).

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The symplectic form is given as $\omega _{\partial M}=\omega _{\Sigma _l}+\omega _{\Sigma _r}$ where

$$\begin{equation} \fl \omega _{\Sigma _{l,r}}(\eta _1,\eta _2) =\frac{\epsilon _{l,r}}{2}\int {\rm d}x_0\, (\eta _1(x_0,x_1) \partial _{x_1} \eta _2(x_0,x_1) - \eta _2(x_0,x_1)\partial _{x_1}\eta _1(x_0,x_1))|_{\Sigma _{l,r}} \end{equation} \tag{ 56 }$$

and _l = −1 and _r = 1. Since the boundary of M is a disjoint union ∂M = Σ_l∪Σ_r the complex structure J_∂M acts as a complex structure $J_{\Sigma _l}$ on $L_{\Sigma _l}$ and $J_{\Sigma _r}$ on $L_{\Sigma _r}$, respectively.

Let us consider a detector at rest at x₁ = 0, i.e., γ(τ) = (τ, 0). Consider the sum P^{Ω, tot}(g → e) := P^{Ω, out}(g → e) + P^{Ω, in}(g → e), that is, the transition probability of a detector given an undetermined one particle state on the boundary. In the canonical theory we know this to be (55). Note that (55) depends on the distance ζ from the origin to the boundary of the regions I and III and on the refraction index n, in other words on information about the physics outside of M. However, P^{Ω, tot}(g → e) does not depend on anything but the identification of the one particle sector and the complex structure. Thus, if we accept that the restriction to the one particle sector in the GBF and the canonical case should encode the same physical situation, we already see that the only way to recover the canonical result is by selecting an appropriate complex structure which encodes this physical information. In particular the correct complex structure for a space time region cannot be given without knowledge of the physics in the spacetime into which the region in question is embedded. In fact it is a priori unclear that it is possible to find such a complex structure.

We will now show that in the particular physical configuration we are considering in this section it is indeed possible to recover the canonical result in equation (55) by an appropriate choice of complex structure for the boundary of the region M. As before, the complex structures J_l and J_r can be specified by giving the corresponding eigenfunctions $e_{p_0;l}{\rm e}^{{\rm i}p_0 x_0}\in L^{\mathbb {C}}_{\Sigma _l}$ and $e_{p_0;\Sigma _R}{\rm e}^{{\rm i}p_0 x_0}\in L^{\mathbb {C}}_{\Sigma _r}$ such that $J_l e_{p_0;l}{\rm e}^{{\rm i}p_0 x_0}={\rm i}e_{p_0;l}{\rm e}^{{\rm i}p_0 x_0}$ and $J_r e_{p_0;r}{\rm e}^{{\rm i}p_0 x_0}={\rm i}e_{p_0;r}{\rm e}^{{\rm i}p_0 x_0}$.

We will show that the complex structure given by setting $e_{p_0;l}(x_1):=c_{a;l}(p_0)\cosh (p_1 x_1)+c_{b;l}(p_0)\sinh (p_1 x_1)$ with the coefficients

$$\begin{eqnarray} \nonumber c_{a;l}(p_0)&=&\cosh (p_1\zeta )+{\rm i}\frac{p}{\sqrt{n}p_1}\sinh (p_1\zeta )\\ c_{b;l}(p_0)&=&\sinh (p_1\zeta )+{\rm i}\frac{p}{\sqrt{n}p_1}\cosh (p_1\zeta ) , \end{eqnarray} \tag{ 57 }$$

and by setting $e_{p_0;r}(x_1):=c_{a;r}(p_0)\cosh (p_1 x_1)+c_{b;r}(p_0)\sinh (p_1 x_1)$ with

$$\begin{eqnarray} \nonumber c_{a;r}(p_0)&=&\cosh (p_1\zeta )-{\rm i}\frac{p}{\sqrt{n}p_1}\sinh (p_1\zeta )\\ c_{b;_r}(p_0)&=&-\sinh (p_1\zeta )+{\rm i}\frac{p}{\sqrt{n}p_1}\cosh (p_1\zeta ) , \end{eqnarray} \tag{ 58 }$$

indeed reproduces the canonical result (55) from equation (62). In particular, this complex structure is non-unitary.

We obtain the probability for the transition of the detector from the ground state to the excited state by summing the expression in equation (35) over a set of mutually orthogonal, normalized states spanning the set of possible states of the scalar field, i.e., by summing over the set representing our ignorance of the final state of the system.

As in the canonical case, we assume that there are no field particles going in the region M. In the canonical case, this was done by fixing the initial state on the initial hypersurface to be the vacuum state. This is more complicated in the case of a timelike hypersurface. However, we can use the transition probability of the detector itself to decide which states correspond to ingoing and outgoing particles by comparing the result to the canonical case: the transition probability from the ground state to the excited state of the detector was zero in the canonical case. From equation (35) with γ(τ) = (τ, 0), we obtain that the one particle states $a^\dagger _{\xi }\psi _{M;0}$ in $\mathcal {H}_{\partial M}$ that are given such that $\hat{\xi }\propto {\rm e}^{{\rm i}p_0x_0}$ with p₀ > 0 and $\hat{\xi }$ derived from ξ as defined in equation (30) lead to a vanishing transition probability. Hence, they are the outgoing states. Conversely, the ingoing one particle states are linear combinations of the states $a^\dagger _{\xi }\psi _{M;0}$ given such that $\hat{\xi }\propto {\rm e}^{{\rm i}p_0x_0}$ with p₀ < 0.

We obtain that for two one particle states we have

$$\begin{equation} \langle a^{\dagger }_\xi \psi _{M;0}, a^{\dagger }_{\xi ^{\prime }} \psi _{M;0}\rangle =\lbrace \xi ^{\prime },\xi \rbrace _{\partial M}=g_{\partial M}(\hat{\xi }^{\prime },\overline{\hat{\xi }}) , \end{equation} \tag{ 59 }$$

where the symmetric, bilinear map g_∂M is extended to $L^{\mathbb {C}}_{\partial M}$ as g_∂M(ϕ + iϕ', ψ) := g_∂M(ϕ, ψ) + ig_∂M(ϕ', ψ) for real ϕ, ϕ', ψ. Hence, two one particle states corresponding to ξ and ξ' are orthogonal if and only if the corresponding $\hat{\xi }$ and $\hat{\xi }^{\prime }$ are orthogonal with respect to g_∂M.

The complex structure J_∂M defined in equations (57) and (58) leaves the energy eigenspaces

$$\begin{eqnarray*} &&L_{M}^{p_0,\mathbb {C}}:=\lbrace c_+ e^+_{p_0;l}+c_- \overline{e^+_{-p_0;l}}| c_+,c_-\in \mathbb {C}\rbrace \end{eqnarray*}$$

invariant. Hence, J_∂M maps ingoing states to ingoing states and outgoing states to outgoing states, and we find from the definition of g_∂M from ω_∂M and J_∂M that ingoing and outgoing states are orthogonal.

Let us now consider just the ingoing states to show that we recover the canonical result in equation (55) for the transition of the detector from its excited state to its ground state: since the complex structure defined by equations (57) and (58) leaves the energy eigenspaces $L_{M}^{p_0,\mathbb {C}}$ invariant, we see from equation (59) that two ingoing states $a^\dagger _{\xi }\psi _{0;M}$ and $a^\dagger _{\xi ^{\prime }}\psi _{0;M}$ are orthogonal if $\hat{\xi }\propto {\rm e}^{{\rm i}p_0x_0}$ and $\hat{\xi }^{\prime }\propto {\rm e}^{{\rm i}p^{\prime }_0x_0}$ with $p_0\ne p^{\prime }_0$. Hence, we obtain an orthonormal set of ingoing one particle states by specifying a set of orthonormal one particle states $a^\dagger _{\xi _1}\psi _{M;0}$ and $a^\dagger _{\xi _2}\psi _{M;0}$ corresponding to solutions ξ₁, ξ₂ in $L_{M}^{p_0,\mathbb {C}}$ for every p₀.

From equation (59) we know that the orthonormality of the one particle states $a^\dagger _{\xi _1}\psi _{0;M}$ and $a^\dagger _{\xi _2}\psi _{0;M}$ is equivalent to the orthonormality of $\hat{\xi }_1$ and $\hat{\xi }_2$ with respect to the sesquilinear map $(\cdot ,\cdot ):=g_{\partial M}(\cdot ,\overline{\cdot })$ which is anti-linear in the second entry. We start with the two linearly independent eigenfunctions of $J_{\Sigma _l}$ defining $\xi ^{+}_{p_0}:=e^+_{p_0;l}$ and $\xi ^{-}_{p_0}:=\overline{\xi ^{+}_{-p_0}}$. We define the normalization constant $c_{p_0}^{+}$ such that $(\xi ^{+}_{p_0},\xi ^{+}_{p^{\prime }_0})=4\pi (c_{p_0}^{+})^{-1}\delta (p_0-p^{\prime }_0)$. An orthonormal basis in the one particle sector is then given by the set $a^\dagger _{\xi _1}\psi _{0;M}$ and $a^\dagger _{\xi _2}\psi _{0;M}$, where

$$\begin{eqnarray} \hat{\xi }_1=\tilde{\xi }^{+}_{p_0}&:=&\sqrt{c_{p_0}^{+}}\xi ^{+}_{p_0} \end{eqnarray} \tag{ 60 }$$

$$\begin{eqnarray} \hat{\xi }_2=\tilde{\xi }^{-}_{p_0}&:=&\sqrt{c_{p_0}^{-}}\left(\xi ^{-}_{p_0}-\frac{(\xi ^{-}_{p_0},\xi ^{+}_{p_0})}{(\xi ^{+}_{p_0},\xi ^{+}_{p_0})}\xi ^{+}_{p_0}\right) , \end{eqnarray} \tag{ 61 }$$

with $\sqrt{c_{p_0}^{-}}$ such that $(\tilde{\xi }^{-}_{p_0},\tilde{\xi }^{-}_{p_0})=4\pi \delta (p_0-p^{\prime }_0)$. From equation (36) we find the corresponding response of the detector as

$$\begin{equation} P^{\Omega ,p_0}(e\rightarrow g)=\frac{\lambda ^2}{m\Omega }\int _{-\infty }^{\infty } {\rm d}\tau \,{\rm d}\tau ^{\prime }\, {\rm e}^{-{\rm i}(\Omega +p_0)(\tau -\tau ^{\prime })} (|\tilde{\xi }^{+}_{p_0}(z)|^2+|\tilde{\xi }^{-}_{p_0}(z)|^2) . \end{equation} \tag{ 62 }$$

Now, we insert the eigenfunctions of the complex structure defined in equations (57) and (58) and recover the expression in equation (55) as it is shown in appendix A.4.

While this is not the only possible complex structure here, it is very instructive to see what happens if we use the equations of motion to unitarily propagate the hypersurface Σ_l and Σ_r and the corresponding complex structures to the regions I and III respectively by considering only the global solution defined in (A.24) which are in one-to-one correspondence with field configurations on Σ_l and Σ_r, respectively⁶.

Employing the boundary conditions in equation (51), we obtain for the global solutions

$$\begin{equation} \fl e^+_{p_0;l}(x_0,x_1)=\left\lbrace \begin{array}{@{}lcr@{}}\Big (e_{p_0;l}(-\zeta )\cos p(x_1+\zeta )&&\\ +\frac{\sqrt{n}}{p}\left.\left(\frac{\rm d}{{\rm d}x_1}e_{p_0;l}(x_1)\right)\right|_{x_1=-\zeta }\sin p(x_1+\zeta )\Big ){\rm e}^{{\rm i}p_0 x_0} & {\rm for} & x_1<-\zeta \\ e_{p_0;l}(x_1)\,{\rm e}^{{\rm i}p_0 x_0} & {\rm for} & -\zeta \le x_1\le \zeta ,\end{array}\right. \end{equation} \tag{ 63 }$$

and

$$\begin{equation} \fl e^+_{p_0;r}(x_0,x_1)=\left\lbrace \begin{array}{@{}lcr@{}}e_{p_0;r}(x_1)\,{\rm e}^{{\rm i}p_0 x_0} & {\rm for} & -\zeta \le x_1\le \zeta \\ \Big (e_{p_0;r}(\zeta )\cos p(x_1-\zeta )&&\\ +\frac{\sqrt{n}}{p} \left.\left(\frac{\rm d}{{\rm d}x_1}e_{p_0;r}(x_1)\right)\right|_{x_1=\zeta }\sin p(x_1-\zeta )\Big )\,{\rm e}^{{\rm i}p_0 x_0} &{\rm for}& \zeta < x_1 .\end{array}\right. \end{equation} \tag{ 64 }$$

Note also that, as in the canonical theory, the extensions to the regions I and III of $e^+_{p_0;l}$ and $e^+_{p_0;r}$ respectively are only continuous, but not analytic at x₁ = ±ζ since the spacetime metric g has a discontinuity at −ζ respectively ζ. Thus in particular an arbitrarily small region in spacetime will no longer automatically contain the complete information on all spacetime without recurse to the equations of motion. Mathematically it is this fact that allows us to study the dependence of the complex structure on the physics outside of the spacetime region.

Through equations (63), (57), (64) and (58) respectively we see that the corresponding unitarily propagated complex structure in region I and III coincides with the complex structure given in equation (46),

$$\begin{equation} J_{\Sigma _l}=-J_{\Sigma _r}=\frac{\partial _{x_1}}{\sqrt{-\partial ^2_{x_1}}} . \end{equation} \tag{ 65 }$$

This is just the complex structure that we found to be the natural one for propagating waves in section 4.1. Thus the complex structure we gave above for evanescent waves is directly related to a natural complex structure for the propagating waves that they turn into asymptotically. This further confirms our interpretation of the complex structure (57), (58) as the correct one for these evanescent modes, and thus that the complex structure, and therefore the vacuum state, on timelike boundaries can depend on the physics outside of the space time region they bound.

In this appendix we briefly review the notion of coherent states for the holomorphic Klein–Gordon field. Fixing any element ξ ∈ L_Σ, the corresponding coherent state $K_{\Sigma ;\xi }\in \mathcal {H}_{\Sigma }$ is the holomorphic function

$$\begin{equation} K_{\Sigma ;\xi }(\phi ) := \exp \left(\case{1}{2}\lbrace \xi ,\phi \rbrace _{\Sigma }\right)\qquad \forall \phi \in L_{\Sigma } . \end{equation} \tag{ A.1 }$$

Note that if we reverse the orientation of Σ, $\Sigma \rightarrow \overline{\Sigma }$, then the symplectic form is mapped to $\omega _{\overline{\Sigma }} = - \omega _{\Sigma }$ and $g_{\overline{\Sigma }} = g_{\Sigma }$ and therefore $J_{\overline{\Sigma }}=-J_\Sigma$ to preserve the positivity. Then we obtain that

$$\begin{equation} \overline{\lbrace \xi ,\phi \rbrace }_{\Sigma } = g_\Sigma (\xi ,\phi ) - {\rm i} \omega _\Sigma (\xi ,\phi ) = g_{\overline{\Sigma }}(\xi ,\phi ) + {\rm i} \omega _{\overline{\Sigma }}(\xi ,\phi ) = \lbrace \xi ,\phi \rbrace _{\overline{\Sigma }} , \end{equation} \tag{ A.2 }$$

and, thus, K_{Σ; ξ} is mapped to $K_{\overline{\Sigma };\xi }=\overline{K_{\Sigma ;\xi }}$. This justifies the identification of the Hilbert spaces $\mathcal {H}_{\Sigma }$ and $\mathcal {H}_{\overline{\Sigma }}$ by the map ι in (32).

Remark that if Σ decomposes into two components, $\Sigma = \Sigma _1 \cup \bar{\Sigma }_2$, then for any $\xi \in L_{\Sigma _1}$ and $\xi ^{\prime } \in L_{\Sigma _2}$, we have

$$\begin{equation} K_{\Sigma ;(\xi ,\xi ^{\prime })} = K_{\Sigma _1;\xi }\otimes K_{\overline{\Sigma _2};\xi ^{\prime }}\in \mathcal {H}_{\Sigma } . \end{equation} \tag{ A.3 }$$

With respect to the inner product defined above, a coherent state has the reproducing kernel property $\langle K_{\Sigma ;\xi },\psi \rangle _{\Sigma } = \psi (\xi )\;\forall \psi \in \mathcal {H}_{\Sigma }$. We define τ^R, τ^I ∈ L_M such that τ = τ^R + J_∂Mτ^I which is always uniquely possible since L_∂M = L_M⊕J_∂ML_M, see proposition 4.2 in [Oec12b]. Then, we obtain for the amplitude

$$\begin{equation} \rho _M(K_{\Sigma ;\tau }) = \exp \left(\frac{1}{4}g_{\partial M}(\tau ^R,\tau ^R) - \frac{1}{4}g_{\partial M}(\tau ^I,\tau ^I)-\frac{{\rm i}}{2}g_{\partial M}(\tau ^R,\tau ^I)\right) . \end{equation} \tag{ A.4 }$$

There is a distinguished coherent state on the hypersurface Σ,

$$\begin{equation} \psi _{\Sigma ;0}:=K_{\Sigma ;0} = 1 \end{equation} \tag{ A.5 }$$

which is the coherent state associated with the vector 0 ∈ L_Σ. This state has a natural interpretation as the vacuum state on Σ. In particular, it satisfies all vacuum axioms posed in [Oec12b]. Note that while the vacuum state is uniquely determined here, its actual functional form depends on the complex structure. The problem of defining a vacuum state has thus been moved into determining the complex structure, where it can be stated as the problem of choosing the positive frequency modes. For more on the equivalence between the two perspectives on the problem of the vacuum see [Oec12e].

In this section we will develop the standard perturbation theory for the GBF in terms of coherent states. and use this to explicitly write the first order coupling of a particle detector.

As detector we will use the model of the harmonic oscillator we developed in section 2. In order to study its coupling to the free scalar field, we will introduce interaction terms as Weyl observables into the free amplitudes. We then develop the perturbation theory for these Interaction terms to first order. This all happens in strict analogy to the standard quantum field theoretic treatment, though our formalism and context are conceptually more general.

We assume that the detector moves on a worldline $\gamma _0:M=[\tau _i,\tau _f]\subset \mathbb {R}\rightarrow \mathcal {M}$ where $\mathcal {M}$ represents spacetime. The boundary of M is then $\partial M=\lbrace \tau _i\rbrace \cup \overline{\lbrace \tau _f\rbrace }$. To recover the standard quantization for the harmonic oscillator we use as in section 2 the complex structure J_τ with

$$\begin{equation} J_{\tau }=-\frac{1}{\Omega }\frac{\rm d}{{\rm d}\tau } . \end{equation} \tag{ A.6 }$$

We model the interaction of the harmonic oscillator q with the field ϕ by considering the classical action

$$\begin{equation*} S_{\mathcal {V}_{UD}}[\phi ,q]=S_{0;1}[\phi ]+S_{0;2}[q]+\mathcal {V}_{UD}(\phi ,q) , \end{equation*}$$

where S_{0; 1}[ϕ] and S_{0; 2}[q] are the free actions for the field and the harmonic oscillator, respectively, and the interaction term is given as [LH07]

$$\begin{equation} \fl \mathcal {V}_{UD}(\phi ,q)=\lambda \int {\rm d}^2x\,\int {\rm d}\tau \, \delta ^{(2)}(x-\gamma (\tau ))\phi (x)q(\tau )=\lambda \int {\rm d}\tau \phi (\gamma (\tau ))q(\tau ) . \end{equation} \tag{ A.7 }$$

Let us express the amplitude for the interacting theory using the path integral:

$$\begin{equation} \rho ^{\mathcal {V}_{{\rm UD}}}(\psi )=\int \mathcal {D}\phi \mathcal {D}q\, \psi (\phi ,q) \,{\rm e}^{{\rm i}S_{\mathcal {V}_{{\rm UD}}}[\phi ,q]} . \end{equation} \tag{ A.8 }$$

Let $\mathcal {H}_1$ be the Hilbert space for the free quantum field ϕ and $\mathcal {H}_2$ be the Hilbert space for the harmonic oscillator q. We define the actions

$$\begin{eqnarray} &&S_{\mu _1}[\phi ]=S_{0;1}[\phi ]+\int {\rm d}^2x \mu _1(x)\phi (x) \end{eqnarray} \tag{ A.9 }$$

$$\begin{eqnarray} &&S_{\mu _2}[q]=S_{0;2}[q]+\int {\rm d}\tau \mu _2(\tau )q(\tau ) , \end{eqnarray} \tag{ A.10 }$$

where μ₁ and μ₂ are test functions defined on the domain of ϕ and q, respectively. Then, the expression in equation (A.8) can be rewritten assuming that the state ψ splits as $\psi =\psi _1\otimes \psi _2\in \mathcal {H}_1\otimes \mathcal {H}_2$ as

$$\begin{eqnarray} \rho ^{\mathcal {V}_{UD}}(\psi )&=& {\rm e}^{{\rm i}\mathcal {V}_{UD}\big(-{\rm i}\frac{\delta }{\delta \mu _1},-{\rm i}\frac{\delta }{\delta \mu _2}\big)}\int \mathcal {D}\phi \mathcal {D}q \,\psi (\phi ,q) \,{\rm e}^{{\rm i}(S_{\mu _1}[\phi ]+S_{\mu _2}[q])}\nonumber \\ &=&{\rm e}^{{\rm i}\mathcal {V}_{UD}\big(-{\rm i}\frac{\delta }{\delta \mu _1},-{\rm i}\frac{\delta }{\delta \mu _2}\big)}\rho ^{\mu _1}(\psi _1)\rho ^{\mu _2}(\psi _2) , \end{eqnarray} \tag{ A.11 }$$

where $\rho ^{\mu _1}$ and $\rho ^{\mu _2}$ are the corresponding amplitudes sometimes called generating functionals. The trick to use generating functionals is well known from the standard formulation of QFT. We generalize this here to the GBF by defining the amplitude for the interacting theory in the GBF as [CO08a]

$$\begin{equation} \rho ^{\mathcal {V}_{UD}}_M(\psi ):={\rm e}^{{\rm i}\mathcal {V}_{UD}\big(-{\rm i}\frac{\delta }{\delta \mu _1},-{\rm i}\frac{\delta }{\delta \mu _2}\big)}\rho ^{\mu _1}_M(\psi _1)\rho _{\gamma _0}^{\mu _2}(\psi _2) . \end{equation} \tag{ A.12 }$$

In particular, to calculate the amplitude $\rho _{M}^{\mu _1}$ we can consider the additional term ∫d²xμ₁(x)ϕ(x) in the corresponding action $S_{\mu _1}[\phi ]$ as a linear observable D₁ giving rise to a Weyl observable: W₁(ϕ) = exp (iD₁(ϕ)) = exp (i∫d²xμ₁(x)ϕ(x)). This is again motivated from the path integral formulation. Hence, we can use equation (27) for the observable amplitude of a Weyl observable.

By considering the harmonic oscillator as a zero-dimensional QFT, i.e. a QFT at a point, we can use a similar construction with the Weyl observable W₂(q) = exp (iD₂(q)) = exp (i∫dτμ₂(τ)q(τ)). Then, we obtain for the boundary state $K_q:=K_{(q_1,q_2)}=K_{q_1}\otimes \overline{K_{q_2}}$ with

$$\begin{eqnarray*} &&q_i(\tau )=\frac{1}{\sqrt{4m\Omega }} (a_i{\rm e}^{-{\rm i}\Omega \tau }+\overline{a_i}{\rm e}^{{\rm i}\Omega \tau }) . \end{eqnarray*}$$

which is the parameterization introduced for the harmonic oscillator in section 2.3, using the unitarity of $J_{\partial \gamma _0}$:

$$\begin{eqnarray} &&\fl \rho ^{\mu _2}_{\gamma _0}(K_{q_1}\otimes \overline{K_{q_2}})=\exp \left(\frac{1}{2}\overline{a_1}a_2+\frac{{\rm i}}{\sqrt{4m\Omega }}\int {\rm d}\tau \,\mu _2(\tau ) (a_1{\rm e}^{-{\rm i}\Omega \tau }+\overline{a_2}{\rm e}^{{\rm i}\Omega \tau })\right. \nonumber \\ &&\left.+\frac{{\rm i}}{2}\int {\rm d}\tau \,{\rm d}\tau ^{\prime }\,\mu _2(\tau ) (1-{\rm i}J_{\partial \gamma _0})\eta _{D,2}(\tau ^{\prime })\right) , \end{eqnarray} \tag{ A.13 }$$

with η_{D, 2} the unique element of $J_{\partial \gamma _0}L_{\gamma _0}$ such that $D_2(q)=2\omega _{\partial \gamma _0}(q,\eta _{D,2})$ for all $q\in L_{\gamma _0}$.

Now, we cannot calculate the full expression in equation (A.8), instead, we define the term of perturbative order n for the coherent states K_ξ for the field and $K_{q}=K_{q_1}\otimes \overline{K_{q_2}}$ for the detector, respectively, as

$$\begin{equation} \fl \rho ^{\mathcal {V}_{UD}}_{M;n}(K_{\xi }\otimes K_{q}):=\left.\frac{1}{n!}\left(\mathcal {V}_{UD}\left(-{\rm i}\frac{\delta }{\delta \mu _1},-{\rm i}\frac{\delta }{\delta \mu _2}\right)\right)^n\rho ^{\mu _1}_M(K_{\xi _1})\rho ^{\mu _2}_{\gamma _0}(K_{q})\right|_{\mu _1,\mu _2=0}. \end{equation} \tag{ A.14 }$$

The amplitude in equation (A.14) is the analog of the scattering matrix of nth order in standard QFT.

In particular, we are interested in the first order which is given as

$$\begin{eqnarray*} &&\rho ^{\mathcal {V}_{UD}}_{M;1}(K_{\xi }\otimes K_{q})=\left.\mathcal {V}_{UD}\left(-{\rm i}\frac{\delta }{\delta \mu _1},-{\rm i}\frac{\delta }{\delta \mu _2}\right)\rho ^{\mu _1}_M(K_{\xi })\rho ^{\mu _2}_{\gamma _0}(K_{q})\right|_{\mu _1,\mu _2=0} . \end{eqnarray*}$$

The amplitude $\rho ^{\mu _\phi }_M(K_{\xi })$ is given by equation (27) as

$$\begin{equation} \rho ^{\mu _1}_M(K_{\xi })=\rho _M(K_\xi )\,{\rm e}^{{\rm i}D(\hat{\xi })+\frac{{\rm i}}{2}D((1-{\rm i}J_{\partial M})\eta _D)} \end{equation} \tag{ A.15 }$$

with η_D ∈ L_∂M such that D(ϕ) = 2ω_∂M(ϕ, η_D). Then, with the fact that the only term of first order in the test function μ₁ is $D(\hat{\xi })$, we obtain with the definition of the potential in equation (A.7):

$$\begin{eqnarray*} \fl \rho ^{\mathcal {V}_{UD}}_{M;1}(K_{\xi }\otimes K_{q})&=& -\lambda \rho _{\gamma _0}(K_q) \rho _M(K_{\xi })\int {\rm d}\tau \hat{q}(\tau )\hat{\xi }(\gamma (\tau )) \\ &=& -\lambda \rho _M(K_{\xi })\,{\rm e}^{\frac{1}{2}\overline{a_1}a_2}\frac{1}{\sqrt{4m\Omega }}\int {\rm d}\tau (a_1{\rm e}^{-{\rm i}\Omega \tau }+\overline{a_2}{\rm e}^{{\rm i}\Omega \tau })\hat{\xi }(\gamma (\tau )) . \end{eqnarray*}$$

Expressing one particle states as first derivatives of coherent states, we derive

$$\begin{eqnarray} \fl \rho ^{\mathcal {V}_{UD}}_{M;1}(a^\dagger _\xi \psi _0\otimes a^\dagger _q \psi _0)&=& \left.2\frac{\rm d}{{\rm d}\alpha }\frac{\rm d}{{\rm d}\beta }\rho ^{\mathcal {V}_{UD}}_{M;1}(K_{\alpha \xi }\otimes K_{\beta q})\right|_{\alpha ,\beta =0}\nonumber \\ &=&\frac{\lambda }{\sqrt{m\Omega }}\int _{\tau _1}^{\tau _2} {\rm d}\tau \,(a_1{\rm e}^{-{\rm i}\Omega \tau }+\overline{a_2}{\rm e}^{{\rm i}\Omega \tau })\hat{\xi }(\gamma (\tau )) . \end{eqnarray} \tag{ A.16 }$$

To interpret this amplitude, recall from equation (4) that in the GBF the probability to find the state in some subspace $\mathcal {J}$ with basis ξ_a, given that it was in some subspace $\mathcal {I}$ with basis $\xi ^{\prime }_b$ can be written as

$$\begin{equation} P(\mathcal {J}|\mathcal {I})=\frac{\sum _{\xi _a}\big|\sum _n\rho ^{\mathcal {V}_{UD}}_{M;n}(\xi _a)\big|^2}{\sum _{\xi ^{\prime }_b}\big|\sum _n\rho ^{\mathcal {V}_{UD}}_{M;n}(\xi ^{\prime }_b)\big|^2} . \end{equation} \tag{ A.17 }$$

Let $\psi \in \mathcal {I}$ be a normalized one particle state. Then the probability for the system to be found in the state ψ is given by taking $\mathcal {I}$ the subspace spanned by ψ. To first non-vanishing order in the perturbation theory this is given by

$$\begin{equation} P(\psi |\mathcal {I})=\frac{\big|\rho ^{\mathcal {V}_{UD}}_{M;1}(\psi )\big|^2}{\sum _{\xi ^{\prime }_b}\big|\rho ^{\mathcal {V}_{UD}}_{M;0}(\xi ^{\prime }_b)\big|^2} , \end{equation} \tag{ A.18 }$$

where $\rho ^{\mathcal {V}_{UD}}_{M;0}$ is just the free amplitude and we used that a one particle state is a linear functional and $\rho ^{\mathcal {V}_{UD}}_{M;0}$ is given as a Gaussian integral over an even domain and hence always vanishes for a one particle state which is an odd function of the field. Let $\mathcal {I}$ be the sector of the boundary Hilbert space spanned by the vacuum and the one particle states, that is, $\lbrace c\psi _{\partial M;0}|\,c\in \mathbb {C}\rbrace \oplus L_{\partial M}$ understood as subspaces of $\mathcal {H}_{\partial M}$, i.e. we consider only states that contain at most one particle, we find that

$$\begin{equation} P(\psi |\mathcal {I})= \big|\rho ^{\mathcal {V}_{UD}}_{M;1}(\psi )\big|^2 . \end{equation} \tag{ A.19 }$$

since the free amplitude for a one particle state vanishes and the amplitude for the vacuum state is always one by normalization. From equation (A.19) we know that the absolute square of $\rho ^{\mathcal {V}_{UD}}_{M;1}(a^\dagger _\xi \psi _0\otimes a^\dagger _q \psi _0)$ can be interpreted as a transition probability within the one particle sector.

We are interested in the transition of the detector from the ground state ψ₀ at τ₁ to a normalized, excited state ψ_e(q) at τ₂ and vice versa at boundary field state $a^\dagger _\xi \psi _0$, for some fixed field configuration ξ. Using the definition of one particle states in section 3.2, the normalized, excited state of the harmonic oscillator at a component of the boundary is easily seen to be

$$\begin{equation} \psi _e(q)=p_{q_e}(q)=\frac{1}{\sqrt{2}}\lbrace q_e,q\rbrace , \end{equation} \tag{ A.20 }$$

with

$$\begin{equation} q_e=\frac{1}{\sqrt{2m\Omega }}({\rm e}^{{\rm i}\Omega \tau }+{\rm e}^{-{\rm i}\Omega \tau }) . \end{equation} \tag{ A.21 }$$

In order to study the transition from the ground state at τ₁ to an excited state at τ₂ we set a₁ = 0 and a₂ = 1 in (A.16) to obtain the boundary state ψ₀(q₁)⊗ψ_e(q₂). We then find that the transition described by this boundary together with the one particle field state $a^\dagger _\xi \psi _{M;0}\in \mathcal {H}_{M}$ has the probability

$$\begin{eqnarray} P^{\Omega ,\xi }(g\rightarrow e)&:=&\frac{2\lambda ^2}{m\Omega }\int _{\tau _1}^{\tau _2} {\rm d}\tau \,{\rm d}\tau ^{\prime }\, {\rm e}^{{\rm i}\Omega (\tau -\tau ^{\prime })} \hat{\xi }(\gamma (\tau ))\overline{\hat{\xi }(\gamma (\tau ^{\prime }))} . \end{eqnarray} \tag{ A.22 }$$

With a₁ = 1 and a₂ = 0 we obtain for the transition from the excited to the ground state

$$\begin{eqnarray} P^{\Omega ,\xi }(e\rightarrow g)&:=&\frac{2\lambda ^2}{m\Omega }\int _{\tau _1}^{\tau _2} {\rm d}\tau \,{\rm d}\tau ^{\prime }\, {\rm e}^{-{\rm i}\Omega (\tau -\tau ^{\prime })} \hat{\xi }(\gamma (\tau ))\overline{\hat{\xi }(\gamma (\tau ^{\prime }))} . \end{eqnarray} \tag{ A.23 }$$

Which are precisely the transition probabilities (35) and (36).

The solutions to the Klein–Gordon equation in the setup of section 4.2 are:

$$\begin{equation} \fl \varphi ^+_{p_0}(x_1)=\left\lbrace \begin{array}{@{}lcr@{}}N_{p_0} {\rm e}^{-{\rm i}px_1}& {\rm for} & x_1<-\zeta \\ N_{p_0}{\rm e}^{{\rm i}p\zeta }(\cosh {p_1 (x_1+\zeta )}+B_{p_0}\sinh {p_1 (x_1+\zeta )}) & {\rm for} & -\zeta \le x_1\le \zeta \\ {\rm e}^{-{\rm i}px_1}+C_{p_0} {\rm e}^{{\rm i}px_1} &{\rm for}& \zeta < x_1\end{array}\right. , \end{equation} \tag{ A.24 }$$

with

$$\begin{eqnarray} \nonumber N_{p_0}&=&{\rm e}^{-2{\rm i}p\zeta }\left(\cosh 2p_1\zeta +\frac{1}{2}\left(B_{p_0}+B_{p_0}^{-1}\right)\sinh 2p_1\zeta \right)^{-1}\\ \nonumber B_{p_0}&=&-{\rm i}\frac{p}{\sqrt{n}p_1}\\ C_{p_0}&=&N_{p_0}\frac{1}{2}\left(B_{p_0}-B_{p_0}^{-1}\right)\sinh 2p_1\zeta , \end{eqnarray} \tag{ A.25 }$$

and $p=(np_0^2-m^2)^{1/2}$ and $p_1=(m^2-p_0^2)^{1/2}$. In this section we will show the orthogonality relation we use in section 4.2.

First of all, we show that solutions of different energy are orthogonal in any case. This we see from the time translation invariance of the symplectic structure:

$$\begin{equation*} 0=\partial _0\omega _{x_0}\big(\overline{\phi ^s_{p_0}},\phi ^r_{p^{\prime }_0}\big)={\rm i} (p_0-p^{\prime }_0)\omega _{x_0}\big(\overline{\phi ^s_{p_0}},\phi ^r_{p^{\prime }_0}\big)=(p_0-p^{\prime }_0)\big(\phi ^s_{p_0},\phi ^r_{p^{\prime }_0}\big) \end{equation*}$$

which means for $p_0\ne p^{\prime }_0$ that $(\phi ^s_{p_0},\phi ^r_{p^{\prime }_0})$ must vanish. Now, equation (53) can be derived using the continuity equation ∂_μj^μ(ϕ, ϕ') = 0. We define $j^\mu (\phi ,\phi ^{\prime }):=\sqrt{g}\,g^{\mu \nu }(\phi \partial _\nu \phi ^{\prime }-\phi ^{\prime }\partial _\nu \phi )$, and then

$$\begin{equation} j^0\big(\overline{\phi ^s_{p_0}},\phi ^r_{p^{\prime }_0}\big)={\rm i}\frac{1}{p_0-p^{\prime }_0}\partial _1j^1\big(\overline{\phi ^s_{p_0}},\phi ^r_{p^{\prime }_0}\big) \end{equation} \tag{ A.26 }$$

which leads to

$$\begin{eqnarray*} \omega _{x_0}\big(\overline{\phi ^s_{p_0}},\phi ^r_{p^{\prime }_0}\big)&=&\frac{{\rm i}}{2}\lim _{\sigma \rightarrow \infty } \left.\frac{1}{p_0-p^{\prime }_0}j^1\big(\overline{\phi ^s_{p_0}},\phi ^r_{p^{\prime }_0}\big)(x) \right|_{x_1=-\sigma }^{x_1 = \sigma }\\ &=&\frac{{\rm i}}{2}\lim _{\sigma \rightarrow \infty } \left.\frac{n}{p-p^{\prime }}\frac{p_0+p^{\prime }_0}{p+p^{\prime }}j^1\big(\overline{\phi ^s_{p_0}},\phi ^r_{p^{\prime }_0}\big)(x) \right|_{x_1 = -\sigma }^{x_1=\sigma } . \end{eqnarray*}$$

Then, we plug in the functions given in (A.24) and neglect all terms that come with the factor p + p' since they vanish in the limit σ → 0. Using the identities $\overline{C_{p_0}}N_{p_0}+C_{p_0}\overline{N_{p_0}}=0$ and $|C_{p_0}|^2+|N_{p_0}|^2=1$ and the representation of the delta function

$$\begin{equation*} \lim _{\sigma \rightarrow \infty }\frac{\sin (p-p^{\prime })\sigma }{p-p^{\prime }}=\pi \delta (p-p^{\prime }) \end{equation*}$$

we arrive at

$$\begin{eqnarray*} \omega _{x_0}\big(\overline{\phi ^+_{p_0}},\phi ^-_{p^{\prime }_0}\big)&=&0\\ \omega _{x_0}\big(\overline{\phi ^+_{p_0}},\phi ^+_{p^{\prime }_0}\big)&=&{\rm i}\sqrt{n}2\pi p_0\delta (p-p^{\prime }) \end{eqnarray*}$$

which leads to equation (53).

First, we find that we can express the eigenfunctions of the complex structures of $J_{\Sigma _l}$ and $J_{\Sigma _r}$ in terms of each other by the relation

$$\begin{equation} {{e^+_{p_0,l}}\choose{\overline{e^+_{-p_0,l}}}}=\left(\begin{array}{cc}A(p_0) & B(p_0) \\ \overline{B(p_0)} & \overline{A(p_0)}\end{array}\right){{e^+_{p_0,r}}\choose{\overline{e^+_{-p_0,r}}}}=:T{{e^+_{p_0,r}}\choose{\overline{e^+_{-p_0,r}}}} , \end{equation} \tag{ A.27 }$$

where

$$\begin{eqnarray*} A(p_0)&=&-\frac{{\rm i}}{2}\left(\frac{\sqrt{n}p_1}{p}-\frac{p}{\sqrt{n}p_1}\right)\sinh 2p_1\zeta + \cosh 2p_1 \zeta \\ B(p_0)&=&\frac{{\rm i}}{2}\left(\frac{\sqrt{n}p_1}{p}+\frac{p}{\sqrt{n}p_1}\right)\sinh 2p_1\zeta \end{eqnarray*}$$

and we find that

$$\begin{equation} T^{-1}=\left(\begin{array}{@{}cc@{}}\overline{A(p_0)} & -B(p_0) \\ -\overline{B(p_0)} & A(p_0)\end{array}\right) . \end{equation} \tag{ A.28 }$$

From that we calculate for $\xi ^+_{p_0}= e^+_{p_0,\Sigma _r}$ and $\xi ^-_{p_0}=\overline{e^+_{-p_0,\Sigma _r}}$

$$\begin{eqnarray*} (\xi ^{+}_{p_0},\xi ^{+}_{p^{\prime }_0})&=& g_{\Sigma _l}(\xi ^{+}_{p_0},\overline{\xi ^{+}_{p^{\prime }_0}})+g_{\Sigma _r}(\xi ^{+}_{p_0},\overline{\xi ^{+}_{p^{\prime }_0}})\\ &=&(|A(p^{\prime }_0)|^2+|B(p^{\prime }_0)|^2+1)g_{\Sigma _r}(\xi ^{+}_{p_0},\overline{\xi ^{+}_{p^{\prime }_0}}) \\ &=&(|A(p_0)|^2+|B(p_0)|^2+1)2\pi p\delta (p_0-p^{\prime }_0) \end{eqnarray*}$$

and the same way

$$\begin{eqnarray*} &&\fl (\xi ^{-}_{p_0},\xi ^{+}_{p^{\prime }_0}) = g_{\Sigma _l}(\xi ^{-}_{p_0},\overline{\xi ^{+}_{p^{\prime }_0}})+g_{\Sigma _r}(\xi ^{-}_{p_0},\overline{\xi ^{+}_{p^{\prime }_0}}) \\ &&{\hphantom{\fl (\xi ^{-}_{p_0},\xi ^{+}_{p^{\prime }_0})}}=-2A(p_0)\overline{B(p_0)} g_{\Sigma _r}(\overline{e^+_{-p_0,r}},e^+_{-p_0,r})=-2A(p_0)\overline{B(p_0)}2\pi p\delta (p_0-p^{\prime }_0) \end{eqnarray*}$$

which leads us to

$$\begin{eqnarray*} (c_{p_0}^{+})^{-1}&=&\frac{p}{\sqrt{n}p_1}(|A(p_0)|^2+|B(p_0)|^2+1)\\ &=&\frac{p}{4\sqrt{n}p_1}\left(6-\frac{np_1^2}{p^2}-\frac{p^2}{np_1^2} +\left(\frac{\sqrt{n}p_1}{p}+\frac{p}{\sqrt{n}p_1}\right)^2\cosh 4p_1\zeta \right) . \end{eqnarray*}$$

Furthermore, we define

$$\begin{eqnarray*} &&\fl g_{p_0}:=-2\frac{p}{\sqrt{n}p_1}A(p_0)\overline{B(p_0)} \\ &&{\hphantom{\fl g_{p_0}:}}=\frac{p}{2\sqrt{n}p_1}\left(\frac{\sqrt{n}p_1}{p}+\frac{p}{\sqrt{n}p_1}\right)\sinh 2p_1\zeta \left(\left(\frac{\sqrt{n}p_1}{p}-\frac{p}{\sqrt{n}p_1}\right)\sinh 2p_1\zeta +2{\rm i}\cosh 2p_1\zeta \right) \end{eqnarray*}$$

and obtain

$$\begin{equation} (c_{p_0}^{-})^{-1}=(c_{p_0}^{+})^{-1}(1-(c_{p_0}^{+})^2|g_{p_0}|^2)=2\frac{p}{\sqrt{n}p_1}. \end{equation} \tag{ A.29 }$$

With the definition in equation (60) we have

$$\begin{eqnarray*} \tilde{\xi }^{+}_{p_0}&:=&\sqrt{c_{p_0}^{+}}\xi ^{+}_{p_0}\\ \tilde{\xi }^{-}_{p_0}&:=&\sqrt{c_{p_0}^{-}}(\xi ^{-}_{p_0}-c_{p_0}^{+}g_{p_0}\xi ^{+}_{p_0})\\ &=&\sqrt{\frac{\sqrt{n}p_1}{2p}}\left(\cosh 2p_1\zeta +\frac{{\rm i}}{2}\left( \frac{np_1^2}{p^2}-\frac{p^2}{np_1^2}\right)\sinh 2p_1 \zeta \right)^{-1}\\ &&\times \left(\cosh p_1(x+\zeta )-{\rm i}\frac{p}{\sqrt{n}p_1}\sinh p_1(x+\zeta )\right) \end{eqnarray*}$$

and arrive at

$$\begin{eqnarray*} &&\fl |\tilde{\xi }^{+}_{p_0}|^2+|\tilde{\xi }^{-}_{p_0}|^2=\frac{1}{2}|N_{p_0}|^2\left(\left(\frac{\sqrt{n}p_1}{p}-\frac{p}{\sqrt{n}p_1}\right)+\left(\frac{\sqrt{n}p_1}{p}+\frac{p}{\sqrt{n}p_1}\right)\cosh 2p_1 x_1 \cosh 2p_1\zeta \right) \end{eqnarray*}$$

with

$$\begin{equation} |N_{p_0}|^{-2}=\frac{1}{8}\left(6-\frac{np_1^2}{p^2}-\frac{p^2}{np_1^2} +\left(\frac{\sqrt{n}p_1}{p}+\frac{p}{\sqrt{n}p_1}\right)^2\cosh 4p_1\zeta \right) \end{equation} \tag{ A.30 }$$

which shows that we recover the canonical result.