The split property for quantum field theories in flat and curved spacetimes

Abstract

The split property expresses a strong form of independence of spacelike separated regions in algebraic quantum field theory. In Minkowski spacetime, it can be proved under hypotheses of nuclearity. An expository account is given of nuclearity and the split property, and connections are drawn to the theory of quantum energy inequalities. In addition, a recent proof of the split property for quantum field theory in curved spacetimes is outlined, emphasising the essential ideas.

Appendix: Proof of Theorem 3.2

Recall that $m_0\ge 0$ has been fixed and that $f\in C_0^\infty ({{\mathbb R}})$ is nonnegative, even, with unit integral, and has a Fourier transform that is real, even, nonnegative and bounded from below by

$$\begin{aligned} \hat{f}(u)\ge \varphi (|u|) \end{aligned}$$

(52)

on $[m_0,\infty )$, where $\varphi :[m_0,\infty )\rightarrow {\mathbb R}^+$ is monotone decreasing. Note that nonnegativity of f and $\hat{f}$ implies that $\hat{f}(m_0)\le \hat{f}(0)=1$ and hence $\varphi (u)\le 1$ for all $u\ge m_0$.

Consider a single Klein–Gordon field of mass $m\ge m_0$ on the symmetric Fock space $\mathscr {F}$ over $L^2({\mathbb R}^3,d^3{\varvec{k}}/(2\pi )^3)$, and define the usual annihilation operators $a({\varvec{k}})$ by

$$\begin{aligned} (a({\varvec{k}})\Psi )^{(n)}({\varvec{k}}_1,\ldots ,{\varvec{k}}_n) = \sqrt{n+1}\Psi ^{(n+1)}({\varvec{k}},{\varvec{k}}_1,\ldots ,{\varvec{k}}_n), \end{aligned}$$

(53)

where for $\Psi \in \mathscr {F}$, $\Psi ^{(n)}$ denotes its n-particle component.⁸ Writing $a^\dagger ({\varvec{k}})$ for the adjoint of $a({\varvec{k}})$ as a quadratic form, the canonical commutation relations are

(54)

and the quantum field is given by

$$\begin{aligned} \Phi (x) = \int \frac{d^3{\varvec{k}}}{(2\pi )^3\sqrt{2\omega }} \left( a({\varvec{k}})e^{-ik_a x^a} + a^\dagger ({\varvec{k}}) e^{ik_a x^a}\right) , \end{aligned}$$

(55)

in which $k^a=(\omega ,{\varvec{k}})$ with $\omega =(\Vert {\varvec{k}}\Vert ^2+m^2)^{1/2}$.

The energy density (with respect to the standard time coordinate) is a sum of Wick squares

$$\begin{aligned} \rho _m(x) = \frac{1}{2}\left( {:}(\nabla _0\Phi (x))^2{:} + \sum _{i=1}^3 {:}(\nabla _i\Phi (x))^2{:} + m^2{:}\Phi (x)^2{:}\right) , \end{aligned}$$

(56)

so, again in a quadratic form sense,

$$\begin{aligned} \rho _m(x)&= \int \frac{d^3{\varvec{k}}}{(2\pi )^3}\frac{d^3{\varvec{k}}'}{(2\pi )^3}\frac{1}{4\sqrt{\omega \omega '}} \left\{ (\omega \omega '+{\varvec{k}}\cdot {\varvec{k}}'+m^2)e^{i(k-k')\cdot x}a^\dagger ({\varvec{k}})a({\varvec{k}}') \right. \nonumber \\&\quad \left. - (\omega \omega '+{\varvec{k}}\cdot {\varvec{k}}'-m^2)e^{-i(k+k')\cdot x}a({\varvec{k}})a({\varvec{k}}')\right\} + \text {H.C.}, \end{aligned}$$

(57)

where $\text {H.C.}$ denotes the hermitian conjugate.

Next, choose any smooth, symmetric, nonnegative function $B:{\mathbb R}^3\times {\mathbb R}^3\rightarrow {\mathbb R}$, with compact support obeying

$$\begin{aligned} \text {supp}\,B\subset \{({\varvec{u}},{\varvec{u}}'): \Vert {\varvec{u}}\Vert ,\Vert {\varvec{u}}'\Vert \in [{\textstyle \frac{1}{2}},1];\quad |\theta ({\varvec{u}},{\varvec{u}}')|<\pi /3\}, \end{aligned}$$

(58)

where $\theta ({\varvec{u}},{\varvec{u}}')$ is the angle between the vectors ${\varvec{u}}$, ${\varvec{u}}'$, and normalised so that

$$\begin{aligned} \int \frac{d^3{\varvec{u}}}{(2\pi )^3}\, \frac{d^3{\varvec{u}}'}{(2\pi )^3} B({\varvec{u}},{\varvec{u}}') =1\,. \end{aligned}$$

(59)

The function $C:{\mathbb R}^3\times {\mathbb R}^3\rightarrow {\mathbb R}$

$$\begin{aligned} C({\varvec{u}},{\varvec{u}}') = \int \frac{d^3{\varvec{u}}''}{(2\pi )^3} B({\varvec{u}}'',{\varvec{u}})B({\varvec{u}}'',{\varvec{u}}') \end{aligned}$$

(60)

is then pointwise nonnegative with support obeying

$$\begin{aligned} \text {supp}\,C\subset \{({\varvec{u}},{\varvec{u}}'): \Vert {\varvec{u}}\Vert ,\Vert {\varvec{u}}'\Vert \in [{\textstyle \frac{1}{2}},1] \}. \end{aligned}$$

(61)

We now define, for $\lambda >0$, the vacuum-plus-two-particle superposition

$$\begin{aligned} \Psi _{m,\lambda } = \mathscr {N}_{m,\lambda }\left[ \Omega + \frac{\lambda }{\sqrt{2}}\int \frac{d^3{\varvec{k}}}{(2\pi )^3}\,\frac{d^3{\varvec{k}}'}{(2\pi )^3} b({\varvec{k}},{\varvec{k}}')a^\dagger ({\varvec{k}})a^\dagger ({\varvec{k}}')\Omega \right] , \end{aligned}$$

(62)

where $\Omega \in \mathscr {F}$ is the Fock vacuum vector, $\mathscr {N}_{m,\tau ,\lambda }$ is a normalisation constant and

$$\begin{aligned} b({\varvec{k}},{\varvec{k}}') = \frac{\varphi (2\sqrt{2} m)}{m^3}B({\varvec{k}}/m,{\varvec{k}}'/m). \end{aligned}$$

(63)

That is, $\Psi _{m,\lambda }^{(0)}=\mathscr {N}_{m,\lambda }$, $\Psi _{m,\lambda }^{(2)}({\varvec{k}},{\varvec{k}}') =\mathscr {N}_{m,\lambda } \lambda b({\varvec{k}},{\varvec{k}}')$ and all other components of $\Psi _{m,\lambda }$ vanish. As b is compactly supported, each $\Psi _{m,\lambda }$ is a Hadamard state. The normalisation constant is

$$\begin{aligned} \mathscr {N}_{m,\lambda } = \left( 1+\lambda ^2\varphi (2\sqrt{2} m)^2 \text {Tr}\,C\right) ^{-1/2} \ge \left( 1+\lambda ^2 \text {Tr}\,C\right) ^{-1/2}, \end{aligned}$$

(64)

where we have used $\varphi \le 1$ and employed the short-hand notation

$$\begin{aligned} \text {Tr}\,C = \int \frac{d^3{\varvec{u}}}{(2\pi )^3} C({\varvec{u}},{\varvec{u}}). \end{aligned}$$

(65)

Using the general formulae ${\langle \Omega \mid a({\varvec{k}})a({\varvec{k}}')\Psi \rangle } = \sqrt{2}\Psi ^{(2)}({\varvec{k}},{\varvec{k}}')$ and

$$\begin{aligned} {\langle \Psi \mid a^\dagger ({\varvec{k}})a({\varvec{k}}')\Psi \rangle } = 2\int \frac{d^3{\varvec{k}}''}{(2\pi )^3} \overline{\Psi ^{(2)}({\varvec{k}}'',{\varvec{k}})}\Psi ^{(2)}({\varvec{k}}'',{\varvec{k}}'), \end{aligned}$$

(66)

for vacuum-plus-two-particle superpositions $\Psi$, the expected normal ordered energy density is

$$\begin{aligned}&{\langle \Psi _{m,\lambda }\mid \rho _m(x) \Psi _{m,\lambda }\rangle } \nonumber \\&\quad = |\mathscr {N}_{m,\lambda }|^2\text {Re}\,\int \frac{d^3{\varvec{k}}}{(2\pi )^3}\,\frac{d^3{\varvec{k}}'}{(2\pi )^3} \frac{1}{\sqrt{\omega \omega '}}\left( \lambda ^2 c({\varvec{k}},{\varvec{k}}')(\omega \omega '+{\varvec{k}}\cdot {\varvec{k}}'+m^2)e^{i(k-k')\cdot x}\right. \nonumber \\&\qquad \left. -\frac{\lambda }{\sqrt{2}} b({\varvec{k}},{\varvec{k}}')(\omega \omega '+{\varvec{k}}\cdot {\varvec{k}}'-m^2)e^{-i(k+k')\cdot x} \right) , \end{aligned}$$

(67)

where

$$\begin{aligned} c({\varvec{k}},{\varvec{k}}') = \frac{\varphi (2\sqrt{2} m)^2 }{m^3}C({\varvec{k}}/m,{\varvec{k}}'/m). \end{aligned}$$

(68)

As $\Psi _{m,\lambda }$ is Hadamard, the expectation value is smooth in x and we can therefore average against f(t), using the fact that $\hat{f}$ is real, to find

$$\begin{aligned} \int {\langle \Psi _{m,\lambda }\mid \rho _m(t,{\varvec{0}}) \Psi _{m,\lambda }\rangle }f(t)\,dt&= |\mathscr {N}_{m,\lambda }|^2\int \frac{d^3{\varvec{k}}}{(2\pi )^3}\,\frac{d^3{\varvec{k}}'}{(2\pi )^3} \frac{1}{\sqrt{\omega \omega '}}\nonumber \\&\quad \times \left( \lambda ^2 c({\varvec{k}},{\varvec{k}}')(\omega \omega '+{\varvec{k}}\cdot {\varvec{k}}'+m^2) \hat{f}(\omega '-\omega )\right. \nonumber \\&\quad \left. -\frac{\lambda }{\sqrt{2}} b({\varvec{k}},{\varvec{k}}')(\omega \omega '+{\varvec{k}}\cdot {\varvec{k}}'-m^2)\hat{f}(\omega +\omega ') \right) . \end{aligned}$$

(69)

We now seek an upper bound on this last quantity. First note that, for $({\varvec{k}},{\varvec{k}}')\in \text {supp}\,c$, we have $\omega ,\omega '\in [\sqrt{5}m/2,\sqrt{2}m]$ and

$$\begin{aligned} \frac{1}{\sqrt{\omega \omega '}}\left( \omega \omega '+{\varvec{k}}\cdot {\varvec{k}}'+m^2\right) \le \frac{4m^2}{\sqrt{5}m/2}= \frac{8m}{\sqrt{5}}, \end{aligned}$$

(70)

while if $({\varvec{k}},{\varvec{k}}')\in \text {supp}\,b$ we have

$$\begin{aligned} \frac{1}{\sqrt{\omega \omega '}}\left( \omega \omega '+{\varvec{k}}\cdot {\varvec{k}}'-m^2\right) \ge \frac{1}{m\sqrt{2}} \left( \frac{m^2}{4}+{\varvec{k}}\cdot {\varvec{k}}' \right) \ge \frac{3m}{8\sqrt{2}}. \end{aligned}$$

(71)

Second, because f and $\hat{f}$ are positive, $\hat{f}(\omega '-\omega )\le \hat{f}(0)=1$; furthermore, for $({\varvec{k}},{\varvec{k}}')\in \text {supp}\,b$ (and so, in particular, $m_0<\sqrt{5}m_0\le \omega +\omega '\le 2\sqrt{2}m_0$) we have

$$\begin{aligned} \hat{f}(\omega +\omega ')\ge \varphi (\omega +\omega ')\ge \varphi (2\sqrt{2} m). \end{aligned}$$

(72)

Accordingly, as $\lambda$ and the functions b and c are positive, we obtain the bound

$$\begin{aligned} \text {L.H.S. of~(69)} \le |\mathscr {N}_{m,\lambda }|^2\int \frac{d^3{\varvec{k}}}{(2\pi )^3}\,\frac{d^3{\varvec{k}}'}{(2\pi )^3} \left( \lambda ^2 c({\varvec{k}},{\varvec{k}}')\frac{8m}{\sqrt{5}} -\lambda b({\varvec{k}},{\varvec{k}}')\frac{3m}{16}\varphi (2\sqrt{2} m) \right) ,\nonumber \\ \end{aligned}$$

(73)

the right-hand side of which can be written as $-|\mathscr {N}_{m,\lambda }|^2 P(\lambda ) m^4 \varphi (2\sqrt{2} m)^2$, where

$$\begin{aligned} P(\lambda ) = \frac{3}{16} \lambda -\lambda ^2 \frac{8}{\sqrt{5}} \int \frac{d^3{\varvec{u}}}{(2\pi )^3}\,\frac{d^3{\varvec{u}}'}{(2\pi )^3} C({\varvec{u}},{\varvec{u}}'), \end{aligned}$$

(74)

as follows on inserting the definitions (63), (68) of $b({\varvec{k}},{\varvec{k}}')$ and $c({\varvec{k}},{\varvec{k}}')$ and using the normalisation (59) of B. Now the quadratic $P(\lambda )$ has a positive maximum at some $\lambda _0>0$ (note that P and $\lambda _0$ are independent of m). Defining $\Psi _{m}=\Psi _{m,\lambda _0}$, we therefore obtain

$$\begin{aligned} \int {\langle \Psi _{m}\mid \rho _m(t,{\varvec{0}}) \Psi _{m}\rangle }f(t)\,dt \le -\Gamma m^4 \varphi (2\sqrt{2} m)^2, \end{aligned}$$

(75)

where

$$\begin{aligned} \Gamma = \frac{P(\lambda _0)}{1+\lambda _0^2 \text {Tr}\,C} \end{aligned}$$

(76)

depends only on the function B (and not on m or $\varphi$). This completes the proof of Theorem 3.2.