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I am following along Chapter 2 of Takagi's Vacuum noise and stress induced by uniform accelerator. I am at the point of performing the Rindler-Fulling Quantization of a real scalar field, where you expand $\phi$ in terms of the Rindler modes in the left and right wedges - I am puzzled as to why you completely ignore the contributions to the field in the future and past wedges. Let's specify to dimension 4 to be concrete.

Minkowski space is partitioned into four regions: $$ \text{Right Rindler Wedge:}\ \ \ \mathcal{R}_{+} = \left\{ \ (x^0, x^1, x^2, x^3) \in \mathbb{R}^{4} \ | \ x^1 > |x^0| \right\} \\ \text{Left Rindler Wedge:}\ \ \ \mathcal{R}_{-} = \left\{ \ (x^0, x^1, x^2, x^3) \in \mathbb{R}^{4} \ | \ x^1 < - |x^0| \right\} \\ \text{Future Wedge:}\ \ \ \mathcal{F} = \left\{ \ (x^0, x^1, x^2, x^3) \in \mathbb{R}^{4} \ | \ x^0 \geq |x^1| \right\} \\ \text{Past Wedge:}\ \ \ \mathcal{P} = \left\{ \ (x^0, x^1, x^2, x^3) \in \mathbb{R}^{4} \ | \ x^0 \leq -|x^1| \right\} $$

Recall that Rindler coordinates $(\eta, \xi, x^2, x^3)$ are related to rectangular Minkowski coordinates $(x^0, x^1, x^2, x^3)$ through the transformation: $$ x^0 = \xi \sinh(\eta)\ , \ \ \ \ \ \ x^1 = \xi \cosh(\eta) $$ The coordinates $(\eta,\xi,x^2,x^3)$ only cover $R_{+}$ and $R_{-}$.

One solves the Klein-Gordon equation $(\Box_{x} - m^2)r_{\mathbf{k}}(x) =0 $ for Rindler mode functions $r_{\mathbf{k}}$ (where $\mathbf{k} = (\Omega,k_2,k_3) \in (0,\infty) \times \mathbb{R} \times \mathbb{R}$ are the mode parameters), with the constraint that they are positive-frequency with respect to Rindler time $\eta$, ie. This means that $\frac{\partial}{\partial \eta} r_{\mathbf{k}} = - i \omega r_{\mathbf{k}}$ for some $\omega > 0$ (taking $r_{\mathbf{k}}^{\ast}$ gives you negative-frequency modes).

One finds that you need a separate solution in each of the Rindler wedges: So you have positive-frequency modes $r^{+}_{\mathbf{k}}$ in $\mathcal{R}_{+}$, and positive-frequency modes $r^{-}_{\mathbf{k}}$ in $\mathcal{R}_{-}$. A little more explicitly you find: $$ r^{+}_{\mathbf{k}}(\eta, \xi, x^2, x^3) =\begin{cases} \ f^{+}_{\mathbf{k}}(\xi)\ e^{ - i \Omega \eta + i k_2 x^2 + i k_3 x^3 } \ \ \ \ \ , \ x \in \mathcal{R}_{+} \\ \ 0 \ \ \ \ \ , \ x \in \mathcal{R}_{-} \end{cases} \\ r^{-}_{\mathbf{k}}(\eta, \xi, x^2, x^3) =\begin{cases} \ 0 \ \ \ \ \ , \ x \in \mathcal{R}_{+} \\ \ f^{-}_{\mathbf{k}}(\xi) \ e^{ + i \Omega \eta + i k_2 x^2 + i k_3 x^3 } \ \ \ \ \ , \ x \in \mathcal{R}_{-} \end{cases} $$ Where $f^{\pm}_{\mathbf{k}}(\xi)$ are terrible functions I don't have the bravery to type out here. For the negative-frequency modes you just take the complex conjugates of the above. The combination of all of these modes $\{ r^{+}_{\mathbf{k}}, r^{-}_{\mathbf{k}} , r^{+\ast}_{\mathbf{k}}, r^{-\ast}_{\mathbf{k}} \}$ are complete over $\mathcal{R}_{+} \cup \mathcal{R}_{-}$. So then Takagi expands the field $\phi$ in terms of this portion of Minkowski space: $$ \phi(x) = \int d^3\mathbf{k}\ \left[ r_{\mathbf{k}}^{+}(x) b_{\mathbf{k}}^{(+)} + r_{\mathbf{k}}^{+\ast}(x) b_{\mathbf{k}}^{(+)\dagger} + r_{\mathbf{k}}^{-}(x) b_{\mathbf{k}}^{(-)} + r_{\mathbf{k}}^{-\ast}(x) b_{\mathbf{k}}^{(-)\dagger} \right] $$

My Question: Why you can expand the field over just this subset $\mathcal{R}_{+}\cup \mathcal{R}_{-}$ of Minkowski space? I would think that you need to expand the field over all points in Minkowski space? I am not sure how to phrase this properly, but shouldn't there contributions to the field $\phi$ coming from $\mathcal{F} \cup \mathcal{P}$?

At least, this is what is normally done when you quantize $\phi$ in terms of rectangular Minkowski time ie in terms of plane-waves $\propto e^{\mp i \sqrt{\mathbf{p}^2+m^2} x^0 \pm i \mathbf{p} \cdot \mathbf{x} }$. Here you'd have a valid expansion of the field $\phi(x)$ for all points in Minkowski space including $x \in \mathcal{F} \cup \mathcal{P}$

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There are two things you should observe. First, the union of the two open wedges is a (non-connected) globally hyperbolic spacetime in its own right, so quantization is possible without problems. Secondly, the union of those pair of wedges is a static spacetime with respect to the boost Killing vector field which is timelike exactly inside these regions (it is lightlike on their boundary but it vanishes at the bifurcation surface and it is spacelike in the remaining past and future wedges). The quantization procedure in the right and left wedges relies upon the standard construction of the static vacuum with respect to that notion of time. That static vacuum is a ground state it being the zero eigenvector of the positive Hamiltonian referred to the boost time (with apposite directions in the two wedges). This construction is impossible in the rest of spacetime. Indeed Fulling-Unruh vacuum and its Fock space are only defined for observables inside the said wedges and cannot be extended to the whole Minkowski spacetime (it has too bad singularities on the Killing horizon). So, in a sense you are right on the fact that some contribution is missed from the remaining regions, in fact, this state cannot be extended to those regions as I said. Conversely, Minkowski vacuum is everywhere defined in Minkowki spacetime and Poincare' invariant. It is a ground state (0 eigenvector of the corresponding positive Hamiltonian) with respect every notion of Minkowski time. As you probably know, Minkowski vacuum restricted to the algebra of field observables localized in the left and right wedges appears as a thermal state with respect to the boost notion of time (a KMS state) in view of the so-called Bisognano-Wichmann (Fulling-Sewell) theorem applied to the simplest case of non-interacting fields... However this restriction cannot be represented as a state (density matrix) in the Fock space constructed upon Fulling vacuum and the algebraic notion of state is necessary... Strictly speaking one should say that Fulling-Unruh vacuum does not exist. What exists is just the thermal state, apparently referring to that notion of vacuum state, arising when restricting Minkowski vacuum.

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  • $\begingroup$ Thanks for the great answer. In the literature I have seen the claim "the Rindler modes can be analytically continued to $\mathcal{F}$ and $\mathcal{P}$" (unfortunately, I can't remember a source for this) $\to$ from what I am getting from your post, this is not necessary? Because there does not exist a good notion of time in those regions (generated by $\frac{\partial}{\partial \eta}$)? I am understanding now the expansion $\phi(x)$ is more precisely over the coordinates where the `time' $\eta$ exists, so more like $\phi(\eta,\xi,x^2,x^3)$ in $\mathcal{R}_{+} \cup \mathcal{R}_{-}$ $\endgroup$ – Greg.Paul Aug 1 '18 at 17:42
  • $\begingroup$ Is there any utility in analytically continuing the Rindler modes to $\mathcal{F}$ and $\mathcal{P}$? $\endgroup$ – Greg.Paul Aug 1 '18 at 17:43
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    $\begingroup$ I think it is useful in some computation, I do not remember. Maybe just to prove that Minkowki vacuum is a thermal state in the left and right wedges. Try to consult Wald's little book on black holes and qft in curved spacetime. The problem with the extension of Rindler vacuum is that its short distance singularity is too bad on the Killing horizon... $\endgroup$ – Valter Moretti Aug 1 '18 at 18:38

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