The Reeh-Schlieder Theorem

The Reeh-Schlieder theorem is perhaps the most notorious result in algebraic quantum field theory, simultaneously one of the least intuitive and most fundamental.

Denote {\mathcal{H}_0} the vacuum sector of Hilbert space, which consists of all states that can be created from the vacuum {\Omega} by local field operators. Note that {\mathcal{H}_0} is not necessarily the full Hilbert space {\mathcal{H}}, which can generically contain other superselection sectors. By “local”, we mean that a field operator {\phi\left( x^\mu\right)} is smeared against some smooth function {f} with finite spacetime support to create the smeared operator {\phi_f=\int\mathrm{d}^{D}x f(x)\phi(x)}, where {D} is the spacetime dimension and {x=(t,\mathbf{x})}. The smearing ensures that the states have finite norm and are thus well-defined members of Hilbert space. Hence states of the form

\displaystyle |\psi_\mathbf{f}\rangle=\phi_{f_1}\ldots\phi_{f_n}|\Omega\rangle \ \ \ \ \ (1)

are sufficient to generate {\mathcal{H}_0}, i.e., any state in {\mathcal{H}_0} can be approximated arbitrarily well by linear combinations of {\psi_\mathbf{f}}. This defines the vacuum sector of the theory.

Classically, one formulates the initial data for a theory on a Cauchy (i.e., complete spacelike) hypersurface {\Sigma}. Quantum mechanically, this is the “time-slice axiom” of QFT [1], and encodes the physical expectation that there exists a dynamical law that enables one to compute fields at arbitrary time given the set of fields at some time slice (e.g., if {\Sigma} is taken to be the surface {t=0}, one could take the functions {f} to have support in some open neighborhood {\mathcal{U}} of {\Sigma}, say {|t|<\epsilon} for some small positive {\epsilon\in\mathbb{R}}). (As a side-comment for holography, note that there are no Cauchy slices in AdS, since data can always sneak in from infinity! This follows from the fact that the boundary is timelike rather than null or spacelike. Hence while any spacelike slice does cover the whole space, there nevertheless exist null geodesics from timelike infinite that do not intersect any point thereupon. However, the Cauchy problem can still be made well-posed within the Einstein static universe, which only covers half the spacetime).

Now consider the vacuum sector on {\Sigma}, that is, the space of states (1) with the support of {f_i} in {\mathcal{U}}. The Reeh-Schlieder theorem (RS) is the remarkable statement that even if we restrict to an arbitrarily small open set {\sigma\subset\Sigma}, and the support of the functions {f_i} to the corresponding neighborhood {\mathcal{U}_\sigma} of {\sigma} in spacetime, the states {\psi_\mathbf{f}} still suffice to generate the vacuum sector {\mathcal{H}_0} of the full theory! This is often phrased as the statement that by acting on the vacuum with operators localized within some small region — say, the room in which you’re reading this — we can create the Moon!

Witten’s proof [2] of this is by contradiction. In particular, if RS were false, then there would exist a state {|\chi\rangle} orthogonal to all {|\psi_\mathbf{f}\rangle} with all {f_i} supported in {\mathcal{U}_\sigma},

\displaystyle \langle\chi|\psi_\mathbf{f}\rangle=0~. \ \ \ \ \ (2)

This equality holds for all functions {f_i} iff it holds without smearing, i.e.,

\displaystyle \langle\chi|\phi(x_1)\ldots\phi(x_n)|\Omega\rangle=0~,\qquad\forall x_i\in\mathcal{U}_\sigma~. \ \ \ \ \ (3)

While this second formula is simpler to deal with, the matrix element of a product of such exactly local fields has singularities given by a function of the {x_i}, and hence it must be rigorously interpreted as a distribution as in (2). We will not bother to restate Witten’s elegant proof here; it can be found in section 2.2 of [2]. The basic idea is to show that if (3) holds for all {x_i\in\mathcal{U}_\sigma}, then it actually holds for all {x_i} in Minkowski spacetime {M_D}. But this implies that {\chi} must vanish, by the definition of the vacuum sector (i.e., the states {\psi_\mathbf{f}} are dense in {\mathcal{H}_0}). Hence only the zero vector is orthogonal to all states created from the vacuum by local operators supported in {\mathcal{U}_\sigma}; i.e., such states are also dense in {\mathcal{H}_0}.

A relevant question is to what extent this relies on the full spacetime being Minkowski, as opposed to some more general spacetime (e.g., AdS). Witten’s proof assumes the existence of a Hamiltonian which annihilates the vacuum state, {H|\Omega\rangle=0}, and is bounded from below by 0 (for holomorphicity arguments). (In fact this first assumption can be weakened: one needs only an energy-momentum operator {P^\mu} such that {\exp\{ic\cdot P\}} is a bounded operator on Hilbert space, where {c} is a general {D}-vector, so that {\exp\{ic\cdot P\}|\Omega\rangle} varies holomorphically with the components {c_i}). One then uses the fact that any point in {M_D} can be reached by zig-zagging back and forth in different timelike directions, and hence the points {x_i} can be shifted anywhere outside {\mathcal{U}_\sigma}. Thus the problem in extending RS to arbitrary globally hyperbolic{^{**}} spacetimes {M} is two-fold: in curved spacetimes, there is no natural analog of the vacuum state, and no natural translation generators {P^\mu}. Nonetheless one expects that an analog of RS should apply; see for example Ian Morrison’s proposed adaptation to AdS [3].

{{}^{**}}Note the restriction to globally hyperbolic spacetimes is intended to ensure we can still define a Cauchy surface. As mentioned above, this doesn’t technically hold in AdS, but the spirit of this requirement — namely, ensuring causality, and avoiding pathologies like closed timelike curves and naked singularities — survives intact, so we’ll follow the masses and consider AdS to be an honorary member of this family.

The Reeh-Schlieder theorem appears disturbing, and has deep consequences for the notion of locality in field theory (more on this later). However, there’s an important caveat involving unitarity here, which we can illustrate following the example from Witten [2]. Consider {\varsigma\subset\Sigma} to be the region of spacetime, spacelike separated from {\sigma}, in which we wish to create the Moon. Define the operator {\mathcal{M}\in\mathcal{U}_\varsigma} to have expectation value 0 in states which do not contain the Moon in region {\varsigma}, and 1 in states that do. Hence

\displaystyle \langle\Omega|\mathcal{M}|\Omega\rangle=0~. \ \ \ \ \ (4)

However, RS implies that states in {\mathcal{U}_\sigma} are dense in {\mathcal{H}_0}, i.e., there exists an operator {a\in\mathcal{U}_\sigma} such that {a\Omega} approximates the state in which {\mathcal{U}_\varsigma} contains the Moon arbitrarily well,

\displaystyle \langle a\Omega|\mathcal{M}|a\Omega\rangle= \langle\Omega|a^\dagger\mathcal{M} a|\Omega\rangle=1~. \ \ \ \ \ (5)

Now, since {a^\dagger\in\mathcal{U}_\sigma}, and {\mathcal{M}} is supported in the spacelike separated region {\mathcal{U}_\varsigma}, these operators commute, and hence (5) becomes

\displaystyle \langle\Omega|\mathcal{M} a^\dagger a|\Omega\rangle=1~. \ \ \ \ \ (6)

If {a} were unitary, then the fact that {a^\dagger a=1} would imply a contradiction between (4) and (6). But RS does not guarantee the existence of a unitary operator in {\mathcal{U}_\sigma} that will create the Moon in the (in principle arbitrarily distance) region {\mathcal{U}_\varsigma}, merely that there exists some operator that will do this. And indeed, we conclude from the above that it is not possible to perform a unitary transformation with support in {\mathcal{U}_\sigma} that effects any change in observables in a spacelike separated region {\mathcal{U}_\varsigma}, since such operators would satisfy {\langle a\Omega|\mathcal{M}|a\Omega\rangle=\langle\Omega|\mathcal{M}|\Omega\rangle}. Note that this example highlights an important relationship between causality and unitarity; we shall comment on this again below.

So, it is not possible for a physical (read: unitary) operation to affect measurements in spacelike separated regions. Rather, the takeaway message of RS is that there are correlations in the vacuum between spacelike separated operators, even in free field theory. In the example above, these manifest in the fact that

\displaystyle \langle\Omega|\mathcal{M} a^\dagger a|\Omega\rangle\neq \langle\Omega|\mathcal{M}|\Omega\rangle\langle\Omega|a^\dagger a|\Omega\rangle~, \ \ \ \ \ (7)

which follows from (4) and (6).

Of course, it’s common knowledge that there are no diffeomorphism-invariance local operators in quantum gravity, and even in gauge theory one already runs into trouble trying to define local operators or factorizing the Hilbert space. But RS highlights the fact that, even in free field theory, Hilbert space still doesn’t factorize! This is why the entanglement entropy of a subregion in free field theory is both infinite and universal (that is, the UV divergence is the same in any state as it is in vacuum): the divergence is not a property of any particular state, but of the fact that {\mathcal{H}\neq\mathcal{H}_\mathcal{U}\otimes\mathcal{H}_{\bar{\mathcal{U}}}}.

Note also that the above example highlights the subtle yet crucial distinction between the related concepts of locality and causality in quantum field theory. The latter is defined as the vanishing of commutators outside the lightcone, and remains intact insofar as we cannot use alter the state in a spacelike separated region by acting with any physical operator. But the concept of locality is badly broken: we can localize free-field operators with appropriately chosen smearing functions, but vacuum correlations imply that the corresponding states are only local in an approximate sense.

Thus one must exercise care in speaking about the localization of states, as distinct from operators, in this context. If by “localized state” one means a state created by operators whose support is restricted to some finite subregion {\mathcal{U}_\sigma}, then RS clearly renders this concept untenable, since we can equally-well create this state by acting with operators in {\mathcal{U}_\varsigma} instead (at the cost of unitarity). Similarly, asserting that localized state is one whose expectation value is non-zero only within some finite region {\sigma} for all operators is equivalent to the claim that {\langle\psi|\!\mathcal{O}\!|\psi\rangle=0\;\;\forall\mathcal{O}\in\mathcal{U}_\varsigma}. But this contradicts the theorem, since we can create any state in {\sigma} by acting in the spacelike separated region {\varsigma}, and thereby reproduce these non-zero expectation values beyond the circumscribed region. For example, let {\psi} be a “localized state” with non-zero expectation value for operators in {\mathcal{U}_\sigma},

\displaystyle \langle\psi|\mathcal{O}_\sigma|\psi\rangle\neq0~, \ \ \ \ \ (8)

but vanishing expectation value for all operators in the spacelike separated region {\mathcal{U}_\varsigma},

\displaystyle \langle\psi|\mathcal{O}_\varsigma|\psi\rangle=0~. \ \ \ \ \ (9)

However, RS implies that operators in {\mathcal{U}_\varsigma} suffice to generate the full (vacuum sector of the) Hilbert space of the theory. Hence there exists an operator {b\in\mathcal{U}_\varsigma} such that the state {b\psi} approximates the state with non-vanishing expectation value arbitrarily well, that is:

\displaystyle \langle b\psi|\mathcal{O}_\sigma|b\psi\rangle= \langle\psi|\mathcal{O}_\sigma b^\dagger b|\psi\rangle= \langle\psi|\mathcal{O}_b|\psi\rangle \neq0~, \ \ \ \ \ (10)

where the operator {\mathcal{O}_b\equiv\mathcal{O}_\sigma b^\dagger b} does not belong to {\mathcal{U}_\sigma}, which contradicts the assertion in (9).

The caveat about unitarity notwithstanding, RS has some interesting implications. For example, the entanglement of the vacuum is what allowed [4] to map the support of precursor states to within a given boundary subregion (dual to a Rindler wedge in the bulk), and hence RS may have implications for bulk reconstruction in this context. It also sheds light on the problem of precursors in AdS/CFT, and for the encoding of bulk information within holographic shadows in general: namely, that this acausal information is encoded in non-unitary operators in the CFT. If correct, this implies that only a limited form of complete bulk reconstruction can ever succeed: RS seems to strengthen the claim that the CFT knows about the entire spacetime via entanglement, but any measurement a boundary observer performs will only be sensitive to timelike or null bulk data (since spacelike data is encoded in non-unitary operators, which are not observables). (EDIT 2021-07-19: I’ve learned a bit since writing this post; see [5], or my later post on the black hole interior for more).

References:

[1] R. Haag, “Local Quantum Physics: Fields, Particles, Algebras.” 1992.

[2] E. Witten, “Notes on Some Entanglement Properties of Quantum Field Theory,” arXiv:1803.04993 [hep-th].

[3]  I. A. Morrison, “Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography,” JHEP 05 (2014) 053, arXiv:1403.3426 [hep-th].

[4]  B. Freivogel, R. Jefferson, and L. Kabir, “Precursors, Gauge Invariance, and Quantum Error Correction in AdS/CFT,” JHEP 04 (2016) 119, arXiv:1602.04811 [hep-th].

[5] R. Jefferson, “Comments on black hole interiors and modular inclusions,” SciPost Physics 6 no. 4, (Apr, 2019), arXiv:1811.08900 [hep-th].

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