How to “resolve a state” with respect to a spacelike hypersurface in Minkowski Spacetime QFT?

Consider usual free QFT in Minkowski spacetime. For simplicity let us consider a real scalar field $$\phi$$. Usually quantization is performed with respect to one inertial reference frame. This is is usually done by decomposing $$\phi(t,\mathbf{x})=\int \dfrac{d^3\mathbf{k}}{(2\pi)^3}\dfrac{1}{\sqrt{2\omega_\mathbf{k}}}(a(\mathbf{k})e^{-ikx}+a^\dagger(\mathbf{k})e^{ikx})$$

Then $$a(\mathbf{k}),a^\dagger(\mathbf{k})$$ are the creation and annihilation operators such that $$a(\mathbf{k})|0\rangle =0$$ determines the (unique) Poincare invariant vacuum. By the way, the concrete space of states of a single particle is then $$L^2(\Omega_{m,+})$$ where $$\Omega_{m,+}$$ is the positive mass-shell in momentum space defined by the condition $$p^2 = m^2$$ and $$p^0 > 0$$.

In particular this means we have one (improper) basis $$|p\rangle$$ for the single particle space and $$|p_1,\dots, p_n\rangle$$ for the $$n$$-particle space, which are the usually considered bases in QFT.

Suppose that the state of the field is $$|\psi\rangle$$.

Now consider one arbitrary spacelike Cauchy surface $$\Sigma$$. This is one way to capture "space at fixed time" for observers instantaneously comoving with the normal $$n^a$$.

My question is: what is the state for such observer?

Intuitively I believe we should have one Hilbert space $$\mathscr{H}_\Sigma$$ and one state $$|\psi(\Sigma)\rangle\in \mathscr{H}_\Sigma$$ which is "how $$|\psi\rangle$$ is perceived by such observers" but I don't know really how to make this precise. I think this is somehow connected to decomposing $$\phi$$ into a complete set of solutions $$u_i$$ with "positive frequency boundary conditions on $$\Sigma$$" namely: $$n^a \partial_a u_i|_\Sigma = i\omega_i u_i \quad \omega_i \in [0,+\infty)$$ but I don't know how to formalize if that's the way.

So given one already defined free QFT state $$|\psi\rangle$$ (for instance by using the procedure outlined above), how do we resolve it with respect to a Cauchy spacelike surface in practice?

• @DanYand, I believe that in the Heisenberg picture the analogue would be to expand $|\psi\rangle$ in a basis adapted to the observables on $\Sigma$. I might be wrong though. I mean, this is what is done for instance in the derivation of Hawking radiation isn't it? Apart from the fact that there one is interested in $\mathcal{H}^+\cup \mathcal{I}^+$ which is not spacelike, what one does is expand the state of the field in a basis appropriate for observables on $\mathcal{I}^+$. And by in practice I mean what we do to actually compute things, not in this operational sense. – user1620696 Jan 4 at 11:11

In QFT in the Heisenberg picture, the field operators satisfy the equations of motion. When we write the field operator as $$\phi(x)\sim \sum_n \big(f_n^*(x) a_n+f_n(x)a_n^\dagger\big),$$ it is understood that the functions $$f_n(x)$$ satisfy the classical equations of motion, where the spacetime coordinates are collectively denoted by $$x$$. This generalizes the concept of the "mass shell." (I'll stick with a free field here to avoid any complications due to non-linear terms in the equations of motion.) In other words, each of the mode functions $$f_n(x)$$ is determined by its "initial" conditions on any Cauchy surface. Therefore, the mode operators $$a_n$$ are also defined by data on any Cauchy surface. The key word here is any. The data on a single Cauchy surface is enough information to define the mode operators $$a_n$$ (and $$a_n^\dagger$$), and for that very reason, there is no sense in which these mode operators are associated with the given Cauchy surface.

Given a complete set of mode functions $$f_n(x)$$, which in turn define the mode operators $$a_n$$, we can express any given state-vector as a linear combination of state-vectors having definite numbers of these quanta. (In QFT, the same operator algebra can have unitarily inequivalent Hilbert-space representations, but I'll ignore that technicality here.) In this sense, a given complete set of modes defines a basis for the Hilbert space. But, for the reason explained above, there is no sense in which this basis is associated with any particular Cauchy surface. Rather, the Hilbert-space basis is associated with the given set of mode functions. We can change the basis by chosing different mode functions, thereby implicitly choosing different creation/annihilation operators. The choice of mode functions can be specified on any given Cauchy surface but is not associated with that Cauchy surface, because the same set of solutions of the equations of motion could have also been specified using data on any other Cauchy surface.

In flat spacetime, one complete set of mode functions consists of the plane-wave functions, $$\exp(-i\omega t-i\mathbf{p}\cdot\mathbf{x})$$. The quantity $$\mathbf{p}$$ here corresponds to the "index" $$n$$ that was used above (which was written as a discrete index to simplify the notation). The coefficient $$\omega$$ depends on $$\mathbf{p}$$ in such a way that each of these functions solves the classical equation of motion, so each of these functions can be unambiguously specified using "initial" conditions on any Cauchy surface. We don't have to use a Cauchy surface of constant $$t$$. The preceding paragraphs generalize this idea to other complete sets of mode functions and to curved spacetimes.

On the other hand, the chosen mode functions have positive or negative frequency in a neighborhood of some given Cauchy surface with respect to a given "time" coordinate, then this extra condition can be used to associate a basis for the Hilbert space with that Cauchy surface. (The same mode functions will generally not have purely positive or negative frequency on other Cauchy surfaces in the same foliation.)

In QFT, the ingredients that define a specific model are [1]:

• a C*-algebra or a von Neumann algebra $$A$$ that contains all of the model's observables;

• a spacetime with a prescribed metric tensor;

• a mapping from regions $$O$$ of spacetime to subalgebras $$A(O)\subset A$$, with observables in $$A(O)$$ interpreted as being "localized in $$O$$";

• a representation of $$A$$ as operators on a Hilbert space. Because of the preceding ingredient (observables localized in spacetime), the Heisenberg picture is implicitly assumed.

These ingredients suffice to specify the model, but additional general principles are needed in order to eliminate useless models. Those principles include the time-slice principle, which says that if $$O$$ is any neighborhood of any Cauchy surface, then $$A(O)=A$$. (In classical field theory, this would be analogous to working exclusively with solutions of the equations of motion rather than thinking of the equations of motion as constraints on the set of all conceivable histories.) With the help of a few such general principles (which I won't review here), we can prove the Reeh-Schlieder theorem [2]: in the context of flat spacetime where the energy-momentum operators are well-defined, the RS theorem says that if $$O$$ is bounded, then operators in $$A(O)$$ cannot annihilate the vacuum state or any other state on which the spacetime translation group acts holomorphically.

An individual observer is usually a localized thing that occupies some neighborhood of a timelike worldline. In that case, the operators that would be used to perceive the state $$|\psi\rangle$$ would be observables localized in that neighborhood of the observer's worldline. According to the RS theorem, such operators cannot annihilate the vacuum state in flat spacetime, and this is true independently of any foliation by Cauchy surfaces. So if the focus of the question is on localized observers, then strict creation/annihilation operators are not the right things to consider. We could consider a whole family of localized observers whose worldlines are orthogonal to the given Cauchy surface, but none of those observers have access to strict creation/annihilation operators. All they can do is probe the properties of the state using a restricted set of local operators.

On the other hand, if the focus of the question is on associating a basis for the Hilbert space with a given Cauchy surface, then at least for a free field, we can use that Cauchy surface to specify initial conditions for a complete set of solutions of the classical equations of motion, and then write the general solution of the operator equation of motion as a linear combination of those complex-valued solutions with operator-valued coefficients.

If there is a region of spacetime (or asymptotic limit) in which some of the given solutions have positive frequency and the others have negative frequency with respect to a given time-coordinate, then the corresponding operator-valued coefficients can be interpreted as creation and annihilation operators. In general, though, a solution that has positive frequency in one region (or in one asymptotic) limit will not have strictly positive frequency in another region (or limit), if the coordinate system (and therefore the wave equation) is time-dependent. So, even though the creation and annihilation operators are not localized (not members of any of the local algebras $$A(O)$$ for any bounded region $$O$$), their interpretation as creation and annihilation operators depends on the region/limit being considered.

For example, in the derivation of Hawking radiation from a black hole formed by a collapsing body, two different complete sets of solutions are considered:

• one whose members have positive/negative frequency in the asymptotic past, and the initial state is taken to be the one annihilated by the associated annihilation operators (because it's asymptotically like Minkowski spacetime);

• one whose memebers have postiive/negative frequency in the asymptotic future (and on the future black hole horizon), and the corresponding creation/annihilation operators are the ones associated with Hawking radiation.

More detail is shown in references [3], [4], and [5], which collectively give a relatively thorough picture of how this works.

Even though a localized observer doesn't have access to non-localizable operators like strict creation/annihilation operators, a localized observer does have access to localized operators that act approximately like creation/annihilation operators. (This is why we can build things like photon-detectors in the real world, although they can never be perfectly noiseless because of the RS theorem.) The goodness of this approximation improves with the increasing size of the region to which the "observer" has access. So maybe there is some sense in which a family of localized observers associated with an individual Cauchy surface can approximately define a basis for the Hilbert space; but I don't know if that's ever been quantified.

References:

[1] Haag (1996), Local Quantum Physics (Springer)

[2] Section 2 in Witten, "Notes on Some Entanglement Properties of Quantum Field Theory", http://arxiv.org/abs/1803.04993. By the way, Witten's analysis is formulated in terms of initial data (a state) on an arbitrary Cauchy surface.

[3] Wipf, "Quantum fields near black holes," https://arxiv.org/abs/hep-th/9801025

[4] Traschen, "An introduction to black hole evaporation," https://arxiv.org/abs/gr-qc/0010055

[5] Polchinski, "The black hole information problem," https://arxiv.org/abs/1609.04036

• thanks for the great answer and for the effort to help. What I want to understand is what is the concrete realization of the Hilbert space $\mathscr{H}_\Sigma$ in a free QFT. This should compare to the usual approach on which the one-particle space $\mathscr{H}$ is simply required to carry a unitary representation of the Poincare group and then can concretely be identified as $L^2(\Omega_m)$ where $\Omega_m$ is the mass shell. By the points you make, is $\mathscr{H}_\Sigma$ just the space of solutions positive-frequency with respect to the normal? – user1620696 Jan 15 at 1:51
• @user1620696 I inserted a couple of paragraphs at the top to try to address this more directly. I also deleted my earlier comments, which are replaced by this answer. – Chiral Anomaly Jan 16 at 3:54