# Rindler-Fulling Quantization - Rindler mode expansion of $\phi$: why are we ignoring the Past and Future Wedges?

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 $x \in \mathcal{F} \cup \mathcal{P}$

• 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 $\mathcal{R}_{+} \cup \mathcal{R}_{-}$ – Greg.Paul Aug 1 '18 at 17:42
• Is there any utility in analytically continuing the Rindler modes to $\mathcal{F}$ and $\mathcal{P}$? – Greg.Paul Aug 1 '18 at 17:43